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RD Sharma Class 12 Exercise 21.11 Differential Equation Solutions Maths-Download PDF Online

RD Sharma Class 12 Exercise 21.11 Differential Equation Solutions Maths-Download PDF Online

Updated on Jan 24, 2022 04:19 PM IST

A heavy burden in the form of a high-end syllabus is being put upon the grade 12 students. Even though learning these subjects adds value to them in one or the other form, the easiest way of learning them is not taught everywhere. RD Sharma solution And mathematics has always been a subject that is put into the tough zone by the students. To make mathematics and the concepts like Differential Equations easier, the RD Sharma Class 12th Exercise 21.11 solution book lends a helping hand to the students.

RD Sharma Class 12 Solutions Chapter21 Differential Equation - Other Exercise

Differential Equations Excercise: 21.11

Differential Equations exercise 21.11 question 1

Answer: r=1+13t2
Given: It is given that initial radius of balloon=1 unit. After 3 seconds, radius of balloon=2 units.
To find: We have to find the radius after time t.
Hint: Use surface area of a balloon =4πr2. Then differentiate it to find the radius.
Solution: We have radius of the balloon=1 unit
After 3 seconds, radius of balloon=2 units
Let r be the radius and A be the surface area of the balloon at any time t.
=dAdtt=dAdt=kt where k is constant.
=d(4πr2)dt=kt=8πrdrdt=kt
Integrating on both sides
=8πr22=kt22+C=4πr2=kt22+C(i)
We are given that unit at t=0,r=1 unit
=4π(1)2=k×0+C=C=4π
And t=3,r=2 units

=4π(2)2=k(3)22+C [From (i)] =16π=92k+4π=92k=12π
By cross multiplication we get
9k=24πk=249πk=83π
Now substitute C=4π and k=83π in equation (i)

4πr2=83πt22+4π4πr24π=86πt24π(r21)=86πt2π(r21)=1486πt2
(r21)=1π13πt2r21=13t2r2=1+13t2=1+13t2



Differential Equations exercise 21.11 question 2

Answer: 20log2 years
Given: Population growth rate =5% per year.
To find: Time taken for increase in population.
Hint: The rate of population increases with increase in time i.e. dPdt=r%×P
Solution: Population growth rate =5% per year
Let initial population be P0 and the population after time t be P
Then,
dPdt=5%×P=dPdt=5100×P
=>dPdt=P20
By cross multiplication we get,
=>20dPp=dt
Integrating on both sides
=>20dPP=dt=>20logP=t+C(i)
At t=0 we have P=P0 [Putting t=0 and P=P0 in equation i]
=>20log(P0)=0+C=>C=20logP0
Putting C=20logP0 in equation (i)
=>20logP=t+20log(P0)=>20logP20log(P0)=t=>20logpp0=t
When P=2P0 we get
=20log(2P0P0)=t=20log2
Hence, the required time period is 20log2 years.


Differential Equations exercise 21.11 question 3

Answer: 58 years
Given: Present Population =1,00,000
To find: When the city will have a population of 5,00,000
Hint: The population of city increase in time i.e. dPdtP and then find the equation using integration.
Solution: The present population is 1,00,000 and the population of a city doubled in the past 25 years
Let P be the population at any time t
Then, dPdtP
dPdt=kPdPdt=kdt
Integrating on both sides we get,

=>20dPP=kdt=>logP=kt+C(i)
At t=0 we take P=P0
Then,
=>logP0=k×0+C [Putting t=0 and P=P0 in equation (i)]
=>C=logP0

Putting C=logP0 in equation (i) we get

=>logP=kt+logP0=>logPlogP0=kt=>logPp0=kt(ii)
When P=2P0 at t=25 we have
log(2PoP)=kt=>log2=25k
Putting k=log225 in equation (ii) we get
=log(pp0)=(log225)t
We assume that t1 be the time take for the population to become 5,00,000 from 1,00,000
Then,logg(5,00,0001,00,000)=(log225)t1
=log5=(log225)t1=25log5=log2t1=t1=25(loglog5loglog2)=t1=25(1.6090.6931) [loglog5=1.609,loglog2=0.6931]
=t1=58.08 Years 
Therefore the required time is 58 years (approximate).



Differential Equations exercise 21.11 question 4

Answer: 2log2log(1110) hours 
Given: Present bacteria count=1,00,000
To find: The amount of hours taken for the count to reach 2,00,000
Hint: Use dCdt=λC and then find the value of lambda using integration.
Solution: The bacteria count is 1,00,000 and rate of increase =10 in 2 hours
Let C be the bacteria count at any time t,
Then dCdtC
=>dCdt=λC [Where λ is the constant]
Integrating on both sides
=>dCdt=λC=>dCC=λdt=>logC=λt+logk(i)
Where loglogk is integral constant
At t=0 we have C=1,00,000
Then log(1,00,000)=λ×0+logk [putting t=0 and C=1,00,000 in equation (i)]
=log(1,00,000)=logk(ii)
 At t=2C=1,00,000+1,00,000(10100)=1,10,000
From equation (i) we have,
=log1,10,000=λ×2+logk [Putting C=1,10,000 and t=2]=log1,10,000=2λ+logk(iii)
Now, subtracting equation (ii) from (iii) we have
=log1,10,000log1,00,000=2λ+logklogk=log(1,10,0001,00,000)=2λ=log1110=2λ=λ=12log(1110)
Now, we have to find time t in which the count reaches 2,00,000
 Now C=2,00,000 in equation (i)=log(2,00,000)=λt+logk
Putting value of lambda and logk, we get
=log(2,00,000)=12log(1110)t+log1,00,000=12log(1110)t=log2,00,000log1,00,000
=12log(1110)t=log(2,00,0001,00,000)=12log(1110)t=log2=t=2log2log(1110)
Hence, the time required =2log2log(1110) hours


Differential Equations exercise 21.11 question 5

Answer: P=Rs1822,t=12 years 
Given: Compound interest=6% per annum,
To find: The amount after 10 years for Rs 1000 and the time in which the amount doubles.
Hint: Use compound interest i.e. dPdt=Pr100
Solution: Let P be the principal
=dPdt=pr100=dPP=r100dt
Integrating on both sides we get,
=>dPP=r100dt=>logP=r100×t+C(i)
At t=0, we have initial principal P=P0
=>log(P0)=0+C=>C=logP0
Now substituting value of C in equation (i)
=>logP=r100t+logP0=>logPlogP0=rt100=>log(pp0)=rt100
Now, P0=1000, t=10 years and r=6%
=>log(P1000)=6×10100=>logPlog1000=0.6=>logP=0.6+log1000
=>logP=loge0.6+log1000[loglogex=x]
=>logP=log(e0.6×1000)=>logP=log(1.822×1000)[logloge0.6=1.822]
=>logP=log(1822)=>P=1822[Using anti-logarithm on both sides]
Therefore P=Rs 1822
Thus Rs 1000 will be Rs 1822 in 10 years.
Now let t1 be the time taken to double Rs 1000 then, P=2000, P0=1000, r=6%
From equation (i)
log(pp0)=rt100=>log(20001000)=6t1100=>log2=6t1100
=6t1=100log2 [By cross multiplication]

=6t1=100×0.6931[log2=0.6931]=t1=69.316=t1=11.55 Years 
=12 years (approximately)
Hence, it will take 12 years to double the amount.


Differential Equations exercise 21.11 question 6

Answer: 9 times, 5log10log3
Given: The amount of bacteria triples in 5 hours.
To find: The amount of bacteria present after 10 hours and also the time for the number of bacteria to become 10 times.
Hint: Use compound interest i.e. dPdt=pr100
Solution: Let A be the amount of bacteria present at time t and A0 be the initial amount of bacteria
 Then, dAdtA=dAdt=λA
Where lambda is the integrating constant
Integrating on both sides we get,
=>dAdt=λA=>dAA=λdt=>logA=λt+C(i)
Where C is integral constant
When t=0, we have A=A0
=>logA0=0+C [Putting t=0 and A=A0 in equation (i)]=>C=logA0
Putting the value of C in equation (i)
=>logA=λt+logA0 =>logAlogA0=λt=>log(AA0)=λt(ii)
Since bacteria triples in 5 hours,
 So, A=3A0 and t=5
Putting in equation (i) we get,
=log(4A0A0)=5λ=log3=5λ=λ=log35
Putting λ=log35 in equation (ii) we get,
=log(AA0)=log35t(iii)
Now let A1 be the amount of bacteria present after 10 hours,
 Then, log(A1A0)=log35×10
=log(A1A0)=2log3=log(A1A0)=2×1.0986[loglog3=1.0986]
=log(A1A0)=2.1972=A1A0=e2.1972=A1=A0e2.1972=A1=9A0[e2.1972=9]
Hence, after 10 hours the number of bacteria will be 9 times the original.
Now, let t1 be the time necessary for the bacteria to be 10 times the amount. So, A=10A0
Then equation (iii) is
=log(AA0)=log35×t=log(10A0A0)=log35×t1=loglog10=log35×t1
=5log10=log3×t1=t1=5log10log3
Hence, the time needed for the bacteria to become 10 times the initial amount =5log10log3 hours


Differential Equations exercise 21.11 question 7

Answer: 312500
Given: Population of city in 1990=2,00,000
Population on city in 2000=2,50,000
To find: Population of city in 2010
Hint: Use dPdt=λP and find the value of constant lambda.
Solution: Let P be the population of the city at time t
 Then, dPdtP
=dPdt=λP
Where lambda is the proportionality constant
=dPp=λdt
Integrating on both sides we get,
=>dPP=λdt=>logP=λt+logk(i)
Where k is integral constant
 Whent =1990, we have P=2,00,000=>log2,00,000=λ×1990+logk(ii)
[utting t=0 and A=A0 in equation (i)]
At
t=2000, we have P=2,50,000=>log2,50,000=λ×2000+logk(iii)
Subtracting equation (ii) and (iii)
=>log2,50,000log2,00,000=2000λ1990λ=>log(2,50,0002,00,000)=10λ=>log(54)=10λ
=>λ=110log(54)
Putting the value of lambda in equation (ii) we get,
=>log2,00,000=199010log(54)+logk=>logk=log2,00,000199log(54)
Substituting the value of lambda in equation (ii) we get
=>logP=(110log54)2010+log2,00,000199log54=>logP=201log54+log2,00,000199log54=>logP=log(53)201+log2,00,000log(54)199
=>logP=log[(53)201(54)199]+log2,00,000=>logP=log[(54)201199×2,00,000]=>logP=log[(54)2×2,00,000]
=>logP=log[2516×2,00,000]=>logP=log3,12,500=>P=3,12,500
Thus, the population of city in 2010=3,12,500


Differential Equations exercise 21.11 question 8

Answer: C(x)=0.075x2+2x+100
Given: C(x)=dCdx=2+0.15x and C(0)=100
To find: Total cost function C(x)
Hint: Use integration and then find out the value of integral constant
Solution: Given that
=C(x)=dCdx=2+0.15x=dC=(2+0.15x)dx
Integrating on both sides we get,
=dC=(2+0.15x)dx=>dC=2 dx+0.15xdx=>C=2x+0.15x22+D(i)
We have C=100 when x=0
=>100=2×0+0.15×(0)22+D=>D=100
Putting the value of D in equation (i) we get,
=>C=2x+0.15x22+100=>C(x)=2x+0.075x2+100
Thus the total cost function C(x)=2x+0.075x2+100


Differential Equations exercise 21.11 question 9

Answer: 8.33%
Given: Compound interest = 8% per annum,
To find: The percentage increase over one year.
Hint: Use compound interest i.e. dPdt=pr100
Solution: Let P be the principal
=dPdt=pr100=dPP=r100dt
Integrating on both sides we get,
=>dPP=r100dt=>logP=r100×t+C(i) [Where C is integral constant]
At t=0, we have initial principal P=P0
=>log(P0)=0+C=>C=logP0
Now substituting value of C in equation (i)
=>logP=r100t+logP0=>logPlogP0=rt100=>log(pp0)=rt100(ii)
For t=1 and r=8% in equation (ii)
=>log(pp0)=8100=>log(pp0)=0.08
=>PP0=e0.08=>PP0=1.0833[e0.08=1.0833]
Subtracting 1 on both sides we get
=>PP01=1.08331
=PP0P0×100
=0.0833×100=8.33%
Hence, amount increases by 8.33% in one year.


Differential Equations exercise 21.11 question 10

Answer: i=ER[1e(KL)t]
Given: Ldidt+Ri=E
To find: To prove that i=ER[1e(RL)t]
Hint: Integrate the linear differential equation to find integral factor.
Solution: Ldidt+Ri=E
Dividing by L
=dtdt+RLi=EL
This is a linear differential equation and comparing it with
=dydx+Py=Q
Such that,
=P=RL,Q=EL
We know that
 I.F. =epdt=eRLdt=e(RL)t
So, the solution is given by
i(If)=Q(If)dt+C=>i[e(RL)t]=EL[e(RL)t]dt+C=>i[e(RL)t]=EL×LR[e(RL)t]+C
=>i[e(RL)t]=ER[e(RL)t]+C=>i=ER[e(RL)t]+C[e(RL)t]
 => i=ER+C[e(RL)t]=>i=ER+C[e(RL)t](i)
Initially no current passes through the circuit so,i=0, t=0
=>0=ER+Ce0=>0=ER+C=>C=ER
Substituting values of C in equation (i)
=>i=ERER[e(RL)t]=>i=ER[1e(RL)t]


Differential Equations exercise 21.11 question 11

Answer: 1klog2 where lambda is thee constant of proportionality.
Given: Decaying rate of radium at any time is proportional to its mass at that time.
To find: We have to find the time when the mass will be half of its initial mass.
Hint: The amount of radium at any time t is proportional to its mass i.e. dAdtA
Solution: Given that
dAdtA
=>dAdt=λA [Where ? is the constant of proportionality and minus sign indicates that A decrease with increase in t]
=>dAA=λdt
Integrating on both sides
=>dAA=λdt=>logA=λt+C(i)
Since initial amount of radium is A0 then
=>logA0=λ×0+C=>logA0=C
Putting value of C in equation (i) we get
=>logA=λt+logA0=>logAlogA0=λt=>logAA0=λt1
=>log12=λt1=>log2=λt1=>t1=1λlog2

Hence, the time taken=1λlog2 where lambda is the constant of proportionality.



Differential Equations exercise 21.11 question 12

Answer: 0.04%
Given: Half like of radium=1590 years.
To find: We have to find the percentage of radium decaying in one year.
Hint:T ake dAdt=λt and integrate it to find rate of λ
Solution: Given that
dAdtA=dAdt=λA [Where ?the constant of proportionality and minus sign is indicates ]
=dAA=λdt [that A decrease with increase in t]
Integrating on both sides
=dAA=λdt=logA=λt+C ........(i)
Since initial amount of radium is A0 then
=logA0=λ×0+C=logA0=C
Putting value of C in equation (i) we get
=logA=λt+logA0=logAlogA0=λt=logAA0=λt................(ii)
Given that its half-life is 1590 years so we put A=12A0 and t=1590
Hence, equation (ii) becomes
=>log(AA0)=λt=>log(A02A0)=λ×1590
=>log(12)=λ(1590)=>log2=λ×1590
Taking minus sign on both sides,
=>log2=1590λ=>λ=log21590
Put =λ=log21590 in equation (ii) we get
=>log(AA0)=(log21590)t=>AA0=e(log21590)t(iii)
Putting t=1 in (iii) to find the amount of radium after one year, we get
=AA0=elog21590=>AA0=0.9996=>A=A00.9996
Percentage amount disappeared in one year
=A0AA0×100
=A0A00.9996A0×100
=A0(10.9996)A0×100
=(10.9996)×100
=0.0004×100
=0.04%


Differential Equations exercise 21.11 question 13

Answer: x2+y2=25
Given: The slope of tangent at a point P(x,y)=xy
To find: We have to find the equation of the curve which passes through (3,4)
Hint: First take the slope of the curve and integrate the equation.
Solution: So,dydx=xy
=ydy=xdx
Integrating on both sides we get,
=ydy=xdx=y22=x22+C(i)
Where C is constant
Since the curve passes through the point (3,4) such that (3,4) satisfy equation (i)
=(4)22=322+C
=>162=92+C=>8=92+C=>C=8+92
=>C=16+92=>C=252
Substituting the value of C in equation (i) we get
=>y22=x22+252
=y2=x2+25 [Multiplying by 2 on both sides]
=x2+y2=25
Hence, found.



Differential Equations exercise 21.11 question 14

Answer: 2xy2xy2=0
Given: The differential equation is yxdydx=y2+dydx.
To find: We have to find the equation of the curve which passes through (2,2)
Hint: We will integrate the given differential equation and then we use the points (2,2)
Solution: we have,
=yxdydx=y2+dydx
Integrating on both sides we get,
=>(yxdydx)=(y2+dydx)=>dyyy2=dx1+x
=>dyy(1y)=dx1+x=>[1y11y]dy=dxx+1
=>log|y|log|1y|=log|x+1|+C(i)
Since the curve passing through the point (2,2) it satisfies equation (i)
Then,
log|2|log|12|=log|2+1|+Clog|2|log|1|=log|3|+CC=log|2|log|3|C=log|23|
Putting the value of C in equation (i) we get
log|y|log|1y|=log|x+1|+log|23|log|y1y|=log|2(x+1)3|
Taking antilogarithm on both sides,
y1y=2(x+1)33y=2(x+1)(1y)3y=2(1+xyxy)3y=22x+2y+2xy
3y+2+2x2y2xy=0y+2+2x2xy=02xy2x2y=0
Hence, required equation is found.


Differential Equations exercise 21.11 question 15

Answer:tan(yx)=log(ex)
Given: The angle is (yxyx)
To find: We have to find the equation of the curve which passes through (1,π4)
Hint: First take the slope of the curve i.e.dydx=tanθ and take a linear equation y=vx
Solution:
The slope of the curve is dydx=tanθ
We have,
=θ=[yxyx]
 Then dydx= tan inverse tan {[yxyx]}=dydx=yxyx Let y=vx
Differentiating with respect to x we get
dydx=v+xdvdxxdvdx=dydxv
xdvdx=dydxyx[v=yx]
xdvdx=v [From equation i]
Integrating on both sides
vdv=1xdxtanyx=log|x|+C(ii)
The curve passes through[1,π4] it satisfices equation (ii)
=logπ4=log|1|+C=C=1
 Put C=1,tanyx=log|x|+1=tanyx=log|x|+1=tanyx=log|ex|
Hence, required equation is found.


Differential Equations exercise 21.11 question 16

Answer:cy4=ex/y
Given: The curve is y=f(x). SupposeP(x,y) is a part of the curve.
To find: We have to find the equation of the curve for which the intercept cut off by a tangent on x-axis.
Hint: Use equation of tangent i.e.Yy=dydx(Xx) then solve the differential equation.
Solution: Equation of tangent to the curve at P is
=Yy=dydx(Xx)
Where (X, Y) is the arbitrary point on the tangent
Putting Y=0 we get
=0y=dydx(Xx)=Xx=ydxdy
When cut off by the tangent on the x-axis
=xydxdy=4y
Therefore ydxdy=4yx
=dxdy=x4yy=dydx=yx4y(i)
This is homogeneous differential equation
Putting y=vx and dydx=v+xdvdx in (i)
We get
=v+xdvdx=vxx4vx=vxx(14v)=v+xdvdx=v14v=xdvdx=v14vv
=xdvdx=vv(14v)14v=xdvdx=vv+4v214v=xdvdx=4v214v
=1xdxdv=14v4v2=4dxx=14vv2dv
Integrating on both sides,
=4dxx=14vv2dv=4dxx=1v2dv4vv2dv=4dxx=1v2dv4dvv2
=4logx+logC=1v4logv=4logx+4logv+logC=1v=4log(xv)+logC=1v
Substituting value of v we get
=4log(x×yx)+logC=xy=4logy+logC=xy
=logy4+logC=xy[alogx=logxa]
=log(y4C)=xy=y4C=exy
Hence, required curve is found.


Differential Equations exercise 21.11 question 17

Answer: proved.
Given: Slope at any point =y+2x
To find: We have to show that the equation of curve which pass through the origin isy+2(x+1)=2e2x
Hint: Use the linear differential equation i.e. dydx+Py=Q
Solution: Slope at any point =y+2x
 i.e. dydx=y+2x
=dydxy=2x
It is a linear differential equation comparing it with dydc+Py=Q
P=1,Q=2x
Integrating factor (If)=ePdx
=> I.F. =e(1)dx=> I.F. =ex
Solution of the equation is given by
=y× I.F. =Q( I.F. )dx+C=>yex=(2x)(ex)dx+C=>yex=2(x)(ex)dx+C
=>yex=2[xex(dxdxexdx)dx]+C[Using integration by parts]
=>yex=2(xex+1exdx)+C
=>yex=2(xexex)+C
=>yex=ex(2x2+Cex)=>y=2x2+Cex=>y+2(x+1)=Cex(i)
If it passes through origin
=0+2(0+1)=Ce0=>C=2
Now equation (i) becomes
=y+2(x+1)=2ex


Differential Equations exercise 21.11 question 18

Answer:ye3x=(23x29)e3x+269e3
Given: Tangent makes an angle (2x+3y) with x-axis
To find: We have to show that the equation of curve which pass through (1,2)
Hint: Find the slope of the tangent then solve using linear differential equation.
Solution: Slope of tangent t=tanθ and tangent makes angle (2x+3y)
=>dydx=tan[tan1(2x+3y)]
=>dydx=2x+3y
=>dydx3y=2x
It is a linear differential equation
Comparing it with dydx+Py=Q
=P=3,Q=2x
 If =ePdx=>If=e3dx=>If=e3x
Solution of equation is given by
=y(If)=Q(If)dx+C
=>y(e3x)=2xe3xdx+C
Using integration by parts we get
=>y(e3x)=2[xe3x(dxdxe3xdx)dx]+C=>y(e3x)=2[xe3x313e3xdx]+C
=>y(e3x)=23xe3x23e3x×13+C=>y(e3x)=23xe3x29e3x+C
Taking e3x common on both sides
=>y=23x29+C(i)
If it passes through (1,2)
=>2=23×123+Ce3=>C=269e3
So equation (i) becomes
=>ye3x=(23x29)e3x+269e3


Differential Equations exercise 21.11 question 19

Answer:xy=2
Given: Let P(x,y) be the point of contact of tangent with curvey=f(x) which intercepts on x-axis =2x
To find: We have to show that the equation of curve which pass through (1,2)
Hint: Use equation of tangent at point P(x,y) is Yy=dydx(Xx)
Solution: Equation of tangent at P(x,y) is Yy=dydx(Xx)
Put Y=0
=>y=dydx(Xx)=>Xx=ydxdy=>X=xydxdy
Co-ordinate of B(xydxdy,0)
Given( intercept on x-axis) = 2x
=>xydxdy=2x=>ydxdy=2xx=>ydxdy=x
=>dxx=dyy [separating variables]
On integrating on both sides, we get
=>dxx=dyy=>logx=logy+C(i)
It is passing through (1,2)
=>log1=log2+C=>C=log2
Put C=log2 in equation (i) we get
=>logx=logylog2=>logx1=log(y2)[logx=logx1]
=>log(1x)=log(y2)=>1x=y2
Hence, xy=2 is required for equation of the curve.


Differential Equations exercise 21.11 question 20

Answer:y=xx+1(x+logx1)
Given:x(x+1)dydxy=x(x+1)
To find: We have to show that the curve which satisfies x(x+1)dydxy=x(x+1) and passes through (1,0)
Hint: Use linear differential equation to solve.
Solution:x(x+1)dydxy=x(x+1)
Dividing by x(x+1)
=dydxyx(x+1)=1 [This is a linear differential equation]
Comparing with dydx+Py=Q we get
P=1x(x+1),Q=1
 Now If =ePdx
=e1x(x+1)dx=e(1x1x+1)dx=e(logxlogx+1)
=elog(xx+1)=elog(xx+1)1[elogx=logx1]
=(xx+1)1[elogx=x]
=x+1x
So, the solution is given by
=y(If)=Q(If)dx+C=y×(x+1x)=x+1xdx+C=(x+1x)y=xxdx+1xdx+C
=(x+1x)y=1dx+1xdx+C=(x+1x)y=x+logx+C(i)
Since curve passes through the point (1,0) it satisfies the equation of the curve
=(1+11)0=1+log1+C
=0=1+0+C[log1=0]
=C=1
Substituting value of C in equation (i) we get
=(x+1x)y=x+logx1=y=(xx+1)(x+logx1)
Hence the required equation for the curve is found.


Differential Equations exercise 21.11 question 21

Answer: 9y+4x2=0
Given: Curve passes through (3,4) and given slope of curve =2yx
To find: We have to show that the equation which satisfies the curve.
Hint: Solve of curve =2yx,dydx=2yx, solve this to find the equation of the curve.
Solution: Slope of curve =2yx
=dydx=2yx=dyy=2xdx
Integrating on both sides
=dyy=2dxx=logy=2logx+logC=y=x2C(i)
As it passes through (3,4)
=4=(3)2C=4=9C=C=49
So equation (i) becomes
=y=49x2=9y=4x2=9y+4x2=0


Differential Equations exercise 21.11 question 22

Answer:3(x+3y)=2(1e3x)
Given: Equation of slope x+3y1
To find: We have to show that the equation of the curve which passes through the origin.
Hint: Use linear differential equation to solve i.e. dydx+Py=Q and its solution y×If=(Q×If)dx whereIf=ePdx
Solution: Equation of slope =x+3y1
=dydx=x+3y=1
=dydx3y=x1 [It is linear differential equation]
Comparing with dydx+Py=Q
P=3,Q=x1=If=ePdx=e3dx
=e3dx=e3x
Solution of the given equation is
=y(If)=Q(If)dx+C=y(e3x)=(x1)(x3x)dx+C=ye3x=(x1)e3xdx[ddx(x1)e3xdx]dx+C[using integration by parts]
=ye3x=(x1)(13e3x)(e8x3)dx+C=ye3x=(x13)e3x13e3xdx+C
=ye3x=(x13)e3x13(esx3)+C
Takinge3x common on both sides,
=y=x3+1319+ce3x
=y=x3+319+Ce3x=y=x3+29+Ce3x(i)
As it passes through origin
=0=03+29+Ce3(0)=C=29
Now, substituting in equation (i)
=y=x3+2929e3x=y=3x+22e8x9=9y=3x+22e3x
=9y+3x=2(1e3x)=3(3y+x)=2(1e3x)
Hence, required equation of curve is found.


Differential Equations exercise 21.11 question 23

Answer:y+1=2ex22
Given: slope =x+yx
Hint: x-coordinate is abscissa and y-coordinate is ordinate
Solution:
According to ques,
dydx=x+yxdydx=x(1+y)dy1+y=xdx
Integrating both sides w.r.t x
dy1+y=xdx
log|1+y|=x22+c........(i) [dxx=logx+c]
Since, the curve passes through (0,1)
log|1+1|=022+cc=log2
Putting the value of c in eqn (i)
log|1+y|=x22+log2log|1+y|log2=x22[logmlogn=lognm]
log1+y2=x221+y2=ex22y+1=2ex22



Differential Equations exercise 21.11 question 24

Answer: x2+y2=Cx
Given: A curve is such that the length of the perpendicular from the origin in the tangent at any point P of the curve is equal to the abscissa of P.
To find: The differential equation of the curve y22xydydxx2=0 also we we have to find the curve
Hint: if the differential equation is homogeneous then put y=vx=dydx=v+xdvdx
Solution: we have y22xydydxx2=0
=dydx=y2x22xy
It is a homogeneous differential equation
 Put y=vx=dydx=v+xdvdx
 Now, xdvdx+v=v2x2x22xvx
=xdvdx=v212vv [Taking x common on right hand side]
=xdvdx=v212v22v=xdvdx=v212v
=xdvdx=(v2+1)2v=2vdvv2+1=dxx
Integrating on both sides we get
=2vdvv2+1=dxx=2vdvv2+1=dxx[t=v2+1,dt=2vdv]=dtt=dxx
=logt=logx+logC=log(v2+1)=logx+logC
=log(v2+1)=logcx=v2+1=cx
=(yx)2+1=cx[y=vx,v=yx]
=y2x2+1=Cx=y2+x2x2=Cx
=y2+x2=Cx2x=y2+x2=Cx
Differentiating with respect to x
=2x+2ydydxC=0
=dydx=C2x2y
Let (h,k) be the point where tangent passes through origin and length is equal to h. So, equation of tangent at (h,k) is
=(yk)=(dydx)(h,k)(xh)=yk=(c2h2k)(xh)
=2ky2k2=xC2hxhC+2h2=x(C2h)2ky+2k2hC+2h2=0
=x(C2h)2ky+2(k2+h2)hC=0
=x(C2h)2ky+2(Ch)hC=0[h2+k2=Ch on the curve ]
=x(C2h)2ky++Ch=0
Length of perpendicular as tangent from origin is
=L=[ax1+by1+ca2+b2]
=[(0)(C2h)+(0)(2k)+hC(C2h)2+(2k)2]=ChC2+4h2+4k24Ch
=ChC2+4(h2+k2Ch)=ChC2+4(0)
=ChC2=ChCL=C
Hence, x2+y2=Cx is the required curve.


Differential Equations exercise 21.11 question 25

Answer:y2=4x
Given: Distance between foot of ordinate of the point of contact and the point of intersection of tangent and x-axis =2x
To find: we have to find the equation of curve which passes through the point (1,2)
Hint: Use equation of tangent at P(x,y) is (Yy)=dydx(Xx)
Solution: Equation of tangent at P(x,y)
Put Y=0
=y=dydx(Xx)=ydxdy=Xx=X=xydxdy
Let co-ordinate of B=(xydxdy,0)
Given distance between foot of ordinate of the point of contact and the point of intersection of tangent and x-axis =2x i.e. BC=2x
=(xydxdyx)2+(0)2=2x [Distance formula =(xx1)2+(yy1)2]
=(1)2(ydxdy)2=2x=1(ydxdy)2=2x
=ydxdy=2x=dxx=2dyy [Separating variables]
Integration on both sides we get
=dxx=2dyy
=logx=2logy+logC(i)
It passes through (1,2)
=log1=2log2+logC=0=2log2+logC[log1=0]
=2log2=logC=log(2)2=logC[2logx=logx2]=log14=logC
Using antilogarithm
=C=14
Substituting the value of C in equation (i) we get
=logx=2logy+log(14)=logx=logy2+log(14)
=logx=logy24=x=y24=y2=4x
Hence y2=4x is the equation of the curve.


Differential Equations exercise 21.11 question 26

Answer:x2+y26x7=0
Given: Equation of the normal at point (x,y) on the curve (Yy)=dxdy(Xx)
To find: we have to find the equation of curve which passes through the point (3,0) if the curve contains the point (3,4)
Hint: Use equation of normal at point (x,y) on the curve is (Yy)=dydx(Xx)
Solution: Equation of normal on point (x,y) on the curve is
=Yy=dxdy(Xx)
It is passing through (3,0)
=0y=dxdy(3x)=y=dxdy(3x)=ydy=(3x)dx
Integrating on both sides
=ydy=(3x)dx=ydy=3dxxdx=y22=3xx22+C(i)
It passes through (3,4)
=422=3×3322+C
=162=992+C
=162=1892+C=162=92+C
=C=1692=C=72
Substituting C=72 in equation (i)
=y22=3xx22+72
Multiplying by 2
=y2=6xx2+7
Hence, the required equation is found.


Differential Equations exercise 21.11 question 28

Answer: 1567 years
Given: Radium decomposes at a rate proportional to the quantity of radium present.
To find: The time taken for half the amount of the original amount of radium to decompose.
Hint: The quantity of radium decomposes at the rate of time t i.e. dAdtA then solve using integration.
Solution: Let A be the quantity of bacteria present in the culture at any time t and initial quantity of bacteria is A0 be the initial quantity of radium.
=dAdtA=dAdt=λA [λ is proportional constant]
=dAA=λdt
Integrating on both sides
=dAA=λdt
=logA=λt+C(i)
Now, A=A0 when t=0
=logA0=0+C=C=logA0
Substituting in equation (i)
=logA=λt+logA0=logAlogA0=λt=logAA0=λt(ii)
Given that in 25 years radium decomposes at 1.1%
So,
A=(1001.1)%=98.9%A=0.989A0,t=25
Substituting in equation (ii)
=log0.989A0A0=25λ=λ=125log(0.989)
Now equation (ii) becomes
=logAA0=[125log(0.989)]t
Now A=12A0
=log(A2A)=125log(0.989)t=log(12)=125log(0.989)t
=log(21)=125log(0.989)t=1log(2)=125log(0.989)t
=t=log2×25log(0.989)
=t=0.6931×250.01106[log2=0.6931 And log(0.989)=0.01106]
=t=1567
Hence, time taken for radium to decay to half the original amount is 1567 years.


Differential Equations exercise 21.11 question 29

Answer:

Given: Slope of tangent =x2+y22xy.
To prove: The curves which slope at any point (x,y) is x2+y22xy are in rectangular hyperbola.
Hint: The homogeneous differential equation put y=vxdydx=v+xdvdx
Solution: Shape of tangent =x2+y22xy
=dydx=x2+y22xy
It is a homogeneous equation
Out y=vx
Then,
=v+xdvdx=x2+v2x22vx2=v+xdvdx=x2(1+v2)2vx2=v+xdvdx=1+v22v
=xdvdx=1+v22vv=xdvdx=1+v22v22v=xdvdx=1v22v
=2v1v2=dxx
=(2vv21)=dxx
=2vv21=dxx [Separating variables]
Integrating on both sides
=2v1v2dv=1xdx=1tdt=1xdx[1v2=t,2vdv=dt]
=logt=logx+logC=log(1v2)=logx+logC=logx=log(1v2)logC=logx=log(1v2)c
=logx1=log(1v2)c=log1x=log(1v2)c
Using antilogarithm
=1x=1v2c=1v2=cx=1(yx)2=cx=1y2x2=cx
=x2y2x2=cx=x2y2=cx2x=x2y2=Cx
Hence, x2y2=Cx is the equation for a rectangular hyperbola.


Differential Equations exercise 21.11 question 30

Answer:x+y=ex1
Given: The slope of tangent at each point of a curve is equal to the sum of the co-ordinate of the point i.e. slope of tangent at (x,y)=x+y
To find: The curves that pass through the origin.
Hint: As dydx+Py=Q then it is a linear differential equation.
Solution: Slope of tangent at (x,y)=x+y
=dydx=x+y=dydxy=x
It is a linear differential equation
Comparing it with dydx+Py=Q
=P=1,Q=x
=If=ePdx
=If=e1dx=If=ex
It is a linear differential equation
Comparing it with dydx+Py=Q
=y(If)=Q(If)dx+C=y(ex)=xexdx+C
=yex=xexdx(dxdxexdx)dx+C [using integration by parts]
=yex=xexexdx+C=yex=xex+ex+C
Dividing by ex on both sides
=y=x1+Cex(i)
As it passes through origin (x,y)=(0,0)
=0=01+Ce0=C=1
Substituting C in equation (i)
=y=x1+ex=x+y=ex1
Hence, equation of curve is found


Differential Equations exercise 21.11 question 31

Answer:y+1=2ex22
Given: The slope of tangent at each point of a curve is equal to the sum of abscissa and product of abscissa and ordinate of the point i.e. dydx=x+y
To find: The curves that pass through the point (0,1)
Hint: Use linear differential equation to find the equation of the curve.
Solution: we have dydx=x+y
=dydxxy=x(i)
This is a linear differential equation of the type
=dydxPy=Q
Where P=x,Q=x
=If=epdx=If=exdx=If=ex22
So solution of equation is given by
=yex22=xex22dx+C(ii)
 Let I=xex22dx Let x22=t
Differentiating on both sides with respect to x
=2x2dx=dt=xdx=dt=xdx=dt
=I=etdt=I=et=I=ex22
Substituting value of I in equation (ii) we get
=y(ex22)=ex22+C
Dividing by ex22
=y=1+Cex22(iii)
As curve passes through (0,1)
=1=1+Ce0=C=2
Substituting value of C in equation (iii) we get
=y=1+2ex22=y+1=2ex22
Hence, equation of required curve is found.


Differential Equations exercise 21.11 question 32

Answer:3y=x3+4
Given: The slope of tangent at each point of a curve is equal to the square of the abscissa of the point i.e. slope of tangent at (x,y)=x2,dydx=x+y
To find: The curves that pass through the point (1,1)
Hint: First separate the given equation of slope then integrate to find the required curve.
Solution: we have dydx=x2
=dy=x2dx
Integrate on both sides
=>dy=x2dx=y=x33+C(i)
As it passes through(1,1)
=1=(1)33+C=1=13+C
=C=1+13=C=43
Substituting value of C in equation (i) we get
=y=x33+43=3y=x2+4
Hence, the required curve is found.


Differential Equations exercise 21.11 question 33

Answer:x2y2=a2
Given: The product of the slope and the ordinate is equal to the abscissa i.e. y(slope of tangent)
= x
To find: The equation of the curves that pass through the point (0,a).
Hint: Use y× slope of tangent =x to solve.
Solution: we have y(slope of tangent) = x
=ydydx=x=ydy=xdx
Integrating on both sides we get
=ydy=xdx=y22=x22+C(i)
The curve passes through (0,a)
=a22=022+C=a22=C=C=a22
Substituting in equation (i) we get
=y22=x22+a22
Multiply 2 on both sides
=y2=x2+a2=x2y2=a2
Hence, required equation of curve is found.


Differential Equations exercise 21.11 question 34

Answer:x+ylogy=y
Given: The x-intercept of the tangent line to a curve is equal to the ordinate of the point of contact.
To find: The particular curve through the point (1,1)
Hint: Use equation of tangent i.e. Yy=dydx(Xx) and compare with dxdy+Py=Q
Solution: Let P(x,y) be the point in the curve y=y(x) such that the tangent at P cuts the co-ordinate at A and B. We know that the equation of tangent Yt=dydx(Xx)
Putting Y=0 then
=y=dydx(Xx)=ydxdy+x=X
Co-ordinate of B=(ydxdy+x,0)
Here, x intercept of tangent = y
=ydxdy+x=y
=dxdyxy1
This is a linear differential equation
Now comparing with dxdy+Py=Q
P=1y,Q=1
=If=ePdy=If=e1ydy
=If=elogy=If=1y
So solution of equation is given by
=x(If)=Q(If)dy+C=x(1y)=(1)(1y)dy+C=xy=logy+C(i)
As it passes through the point (1,1)
=11=log1+C=1=C[log1=0]
Substituting value of C in equation (i) we get
=xy=logy+1=x=y(logy+1)=x=yylogy=x+ylogy=y
Hence, required equation of curve is found.


The only set of solution books that most schools have availed is the RD Sharma books. Class 12, mathematics, chapter 21, Differential Equations, contains around eleven exercises. The last exercise, ex 21.11, has about 34 questions, including its subparts. The concepts covered in this exercise include basic ideas on differential equations, solving a first-order equation, forming a differential equation, first degree equations, etc. The RD Sharma Class 12 Chapter 21 Exercise 21.11 reference material is enough to solve this section's doubts. Therefore, the students need not worry regarding the challenges they would face while doing their homework.

As it is the last exercise of the chapter, the students must be aware of the previous exercises' sums to understand this exercise effectively. The students can make use of the other RD Sharma books to revise those exercises and the RD Sharma Class 12th Exercise 21.11 book to refer to the sums in this exercise. Everyone who has scored well in their public exams has admitted that the benefits of the RD Sharma books are ineffable. As many questions were picked from the practice section of this book, students who use it get trained for their public exams directly.

The solutions are given in the Class 12 RD Sharma Chapter 21 Exercise 21.11 Solution book updates according to the latest edition of NCERT books. Therefore, students can work out the Differential equations sums by themselves and later recheck the answer with the help of the RD Sharma Class 12th Exercise 21.11 guide. Else, rechecking the sums would be a bit hectic, and they tend to lose marks because of this.

The best way to own the RD Sharma books is by downloading them from the Career 360 website. Be it the RD Sharma Class 12 Solutions Differential Equation Ex 21.11 or the other books; everything is available for free of cost at this site. Many students have reaped the goodness of this book, and now it's your turn to own a copy of the RD Sharma Class 12 Solutions Chapter 21 Ex 21.11 and prepare for your exams.

RD Sharma Chapter wise Solutions


Frequently Asked Questions (FAQs)

1. Which is the most recommended mathematics guide to refer to the Differential Equations concept?

The RD Sharma Class 12th Exercise 21.11 is the most recommended reference guide for the students to refer to the Differential Equations concept. 

2. How can I own a copy of the RD Sharma books for free?

To get a copy of the RD Sharma books for free of cost, you can download it from the Career 360 website.

3. Is the last exercise of the Differential Equations Chapter complex for the students to learn?

Exercise 21.11 is the last exercise of the Differential Equations chapter. Students can use the RD Sharma Class 12th Exercise 21.11 reference book to work out the sums effortlessly in this exercise.

4. Are the solutions given in the RD Sharma books well verified?

The answer key present in the RD Sharma solution guides is rechecked and verified to assure its accuracy. Therefore, the students need not worry about such issues

5. Do the RD Sharma books provide solutions for Level 2 questions?

The RD Sharma books contain the solved sums for every question in the Level 1, Level 2, MCQ, FBQ, RE, and even CSBQ section asked in the textbook.

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