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Consider a tank with a hole at the bottom. The exit rate of water is a function of the water level in the tank. As the water level reduces, the flow rate reduces. This can be represented with a differential equation.
The NCERT Solutions for Class 12 Maths Chapter 9 – Exercise 9.3 teach students exactly how to determine solutions to such equations. These are designed by subject matter experts from Careers360 as per the new CBSE 2025–26 syllabus and are described in depth in step-by-step manner. NCERT Chapter 9 Exercise 9.3 is differential equations under the category of variable separable, where the variables can be rearranged such that all the terms of one variable are placed on one side and the remaining terms are placed on the other side. Students will also encounter special cases where the equation needs to be transformed into separable form prior to using the method.
This material provides easy solutions to all the questions of Exercise 9.3 of Differential Equations. The PDF can be downloaded by the students for practice and enhancement of the chapter for board and and competitive exams
Question 2: Find the general solution:
Answer:
Given, in the question
The required general solution:
Question 5: Find the general solution:
Answer:
Given, in the question
Let,
This is the general solution
Question 7: Find the general solution:
Answer:
Given,
let logy = t
=> 1/ydy = dt
This is the general solution
Question 8: Find the general solution:
Answer:
Given, in the question
This is the required general equation.
Question 11: Find a particular solution satisfying the given condition:
Answer:
Given, in the question
Now,
Now comparing the coefficients
Solving these:
Putting the values of
Therefore,
Question 12: Find a particular solution satisfying the given condition:
Answer:
Given, in the question
Let,
Now comparing the values of
Solving these:
Now putting the values of
Given,
Therefore,
Question 13: Find a particular solution satisfying the given condition:
Answer:
Given,
Now, y =1 when x =0
1 = 0 + c
Therefore, c = 1
Putting the value of c:
Question 14: Find a particular solution satisfying the given condition:
Answer:
Given,
Now, y=1 when x =0
1 = ksec0
Putting the vlue of k:
y = sec x
Question 15: Find the equation of a curve passing through the point (0, 0) and whose differential equation is
Answer:
We first find the general solution of the given differential equation
Given,
Now, Since the curve passes through (0,0)
y = 0 when x =0
Putting the value of c, we get:
Question 16: For the differential equation
Answer:
We first find the general solution of the given differential equation
Given,
Now, Since the curve passes through (1,-1)
y = -1 when x = 1
Putting the value of C:
Answer:
According to the question,
Now, Since the curve passes through (0,-2).
x =0 and y = -2
Putting the value of c, we get
Answer:
Slope m of line joining (x,y) and (-4,-3) is
According to the question,
Now, Since the curve passes through (-2,1)
x = -2 , y =1
Putting the value of k, we get
Answer:
Volume of a sphere,
Given, Rate of change is constant.
Now, at t=0, r=3 and at t=3 , r =6
Putting these value:
Also,
Putting the value of c and k:
Answer:
Let p be the principal amount and t be the time.
According to question,
Now, at t =0 , p = 100
and at t =10, p = 200
Putting these values,
Also,
,
So value of r = 6.93%
Answer:
Let p be the principal amount and t be the time.
According to question,
Now, at t =0 , p = 1000
Putting these values,
Also, At t=10 \,
After 10 years, the total amount would be Rs.1648
Answer:
Let n be the number of bacteria at any time t.
According to question,
Now, at t=0, n = 100000
Again, at t=2, n= 110000
Using these values, for n= 200000
Also check -
Topics | Description | Example |
Variable Separable Method | Involves separating variables x and y on opposite sides of the equation and integrating both sides. Used when a differential equation can be expressed as: | Solve: |
Equations Already in Separable Form | Some differential equations are already arranged so that dy and dx can be directly separated. | |
Implicit Solutions | Solutions are sometimes left in implicit form (not solved for y explicitly). This is valid and acceptable in many cases. | From above: |
Solving Using Initial Conditions | After finding the general solution, plug in the given values (initial conditions) to find the constant C for the particular solution. | Given: General: Apply: |
Rearranging to Separable Form | If the equation isn't already in separable form, rearrange terms algebraically to isolate variables on each side. | Solve: |
Also Read-
As per latest 2024 syllabus. Maths formulas, equations, & theorems of class 11 & 12th chapters
The differential equations of order 1 and degree 1 are handled.
Differential equations of variable separable type are solved in the NCERT solutions for Class 12 Maths chapter 9 exercise 9.3.
Six example problems and their solutions are given under topic 9.5.1.
Twenty-three questions are explained in the Class 12th Maths chapter 6 exercise 9.3.
One question is of objective type with 4 choices.
Three subtopics are discussed in topic 9.4.
3 methods are discussed.
The topics discussed are listed below
Differential equations of variable separable type, homogeneous and linear differential equations.
Admit Card Date:06 May,2025 - 20 May,2025
Admit Card Date:06 May,2025 - 20 May,2025
Changing from the CBSE board to the Odisha CHSE in Class 12 is generally difficult and often not ideal due to differences in syllabi and examination structures. Most boards, including Odisha CHSE , do not recommend switching in the final year of schooling. It is crucial to consult both CBSE and Odisha CHSE authorities for specific policies, but making such a change earlier is advisable to prevent academic complications.
Hello there! Thanks for reaching out to us at Careers360.
Ah, you're looking for CBSE quarterly question papers for mathematics, right? Those can be super helpful for exam prep.
Unfortunately, CBSE doesn't officially release quarterly papers - they mainly put out sample papers and previous years' board exam papers. But don't worry, there are still some good options to help you practice!
Have you checked out the CBSE sample papers on their official website? Those are usually pretty close to the actual exam format. You could also look into previous years' board exam papers - they're great for getting a feel for the types of questions that might come up.
If you're after more practice material, some textbook publishers release their own mock papers which can be useful too.
Let me know if you need any other tips for your math prep. Good luck with your studies!
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I hope this information helps you.
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If you have uploaded screenshot of your 12th board result taken from CBSE official website,there won,t be a problem with that.If the screenshot that you have uploaded is clear and legible. It should display your name, roll number, marks obtained, and any other relevant details in a readable forma.ALSO, the screenshot clearly show it is from the official CBSE results portal.
hope this helps.
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As per latest 2024 syllabus. Physics formulas, equations, & laws of class 11 & 12th chapters
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