Are these practice questions for Differential Equations based on the CBSE pattern?

Yes. Every practice question is modeled on the CBSE Class 12 - Science Mathematics exam pattern — including MCQs, assertion-reason questions, and short-answer formats that frequently appear in CBSE papers.

Are the practice questions for Differential Equations free?

Yes. Every chapter includes free practice questions for Class 12 - Science students to get you started.

What types of questions are included?

The practice set for Differential Equations mainly includes MCQs and CBSE-style short-answer questions, with instant feedback on each answer. For full-length textbook practice, use the NCERT Solutions page.

Are the practice questions suitable for board exam preparation?

Yes. The questions test the same concepts and depth that CBSE asks in board exams. After finishing, also solve the NCERT and NCERT Exemplar exercises for Differential Equations for full coverage.

How many practice questions are there for Differential Equations?

The number scales with chapter length and concept density. Each chapter typically has 15-50 practice questions across MCQ and short-answer formats.

Differential Equations Class 12 Mathematics Practice Questions with Answers

easy

Identify the order and degree of the differential equation

y'' + (y')^2 + 2y = 0

. Explain your reasoning.

Try solving it in your notebook first, then check the solution.

easy

Define the order of a differential equation.

Try solving it in your notebook first, then check the solution.

easy

Name the type of differential equation represented by

\frac{dy}{dx} + Py = Q

, where P and Q are constants or functions of x only.

Try solving it in your notebook first, then check the solution.

easy

What is a 'particular solution' of a differential equation?

Try solving it in your notebook first, then check the solution.

easy

Solve the differential equation

\frac{dy}{dx} = \frac{1+y^2}{1+x^2}

.

Try solving it in your notebook first, then check the solution.

easy

Critique the statement: "The order and degree of a differential equation are always equal." Justify your conclusion with a counterexample.

Try solving it in your notebook first, then check the solution.

easy

Explain the difference between a general solution and a particular solution of a differential equation.

Try solving it in your notebook first, then check the solution.

easy

Analyze the differential equation

(\frac{d^3y}{dx^3})^2 + 5(\frac{dy}{dx})^4 - \log(x) = 0

to determine its order and degree.

Try solving it in your notebook first, then check the solution.

easy

Identify the order and degree of the differential equation

y' + 5y = 0

.

Try solving it in your notebook first, then check the solution.

medium

Justify why the degree of the differential equation

\frac{d^2y}{dx^2} + \sin\left(\frac{dy}{dx}\right) = 0

is not defined, whereas the degree of the equation

\left(\frac{dy}{dx}\right)^2 + \sin y = 0

is defined.

Try solving it in your notebook first, then check the solution.

medium

Determine the order and degree (if defined) of the differential equation

\frac{d^2y}{dx^2} = \sqrt[3]{1 + (\frac{dy}{dx})^5}

.

Try solving it in your notebook first, then check the solution.

medium

Examine if the function

y = A e^{2x} + B e^{-2x}

is a solution to the differential equation

\frac{d^2y}{dx^2} - 4y = 0

.

Try solving it in your notebook first, then check the solution.

medium

Solve the homogeneous differential equation

y' = \frac{x+y}{x}

.

Try solving it in your notebook first, then check the solution.

medium

Find the equation of a curve that passes through the point

(1, -1)

and whose differential equation is

xy \frac{dy}{dx} = (x+2)(y+2)

.

Try solving it in your notebook first, then check the solution.

medium

The differential equation for a family of curves is given by

\frac{dy}{dx} = \frac{y^2-x^2}{2xy}

. A student solves it and presents the solution as

x^2+y^2 = kx

. Evaluate this proposed solution by checking if it satisfies the differential equation. If it is incorrect, derive the correct solution and justify your steps.

Try solving it in your notebook first, then check the solution.

medium

Describe the steps to find the Integrating Factor (I.F.) for a linear differential equation of the form

\frac{dy}{dx} + Py = Q

.

Try solving it in your notebook first, then check the solution.

medium

Solve the differential equation

\frac{dy}{dx} + 2xy = 2xe^{-x^2}

.

Try solving it in your notebook first, then check the solution.

medium

Calculate the integrating factor (I.F.) for the linear differential equation

x \frac{dy}{dx} - 3y = x^3

.

Try solving it in your notebook first, then check the solution.

medium

Describe the complete procedure to solve a homogeneous differential equation of the form

\frac{dy}{dx} = g\left(\frac{y}{x}\right)

. List all the necessary substitutions and steps.

Try solving it in your notebook first, then check the solution.

medium

A student claims that any homogeneous differential equation of the form

\frac{dy}{dx} = F(\frac{y}{x})

must have a degree of 1. Evaluate this claim and provide a justification for your conclusion.

Try solving it in your notebook first, then check the solution.

medium

An incorrect solution is provided for the differential equation

(x^2 - 3y^2)dx + 2xy dy = 0

. The proposed solution is

x^2 - y^2 = Cx^3

. Critique this solution by checking its validity. Then, formulate the correct approach and derive the correct general solution.

Try solving it in your notebook first, then check the solution.

medium

Find the particular solution of the differential equation

\frac{dy}{dx} - 3y \cot x = \sin 2x

, given that

y=2

when

x = \frac{\pi}{2}

.

Try solving it in your notebook first, then check the solution.

medium

Propose a suitable substitution to reduce the differential equation

\frac{dy}{dx} = \cos(x+y)

to a variable separable form and justify your choice.

Try solving it in your notebook first, then check the solution.

medium

Explain the concept of an 'ordinary differential equation' and a 'partial differential equation'. Provide one example for each to illustrate the difference.

Try solving it in your notebook first, then check the solution.

medium

State the condition for a differential equation of the form

\frac{dy}{dx} = F(x, y)

to be classified as 'homogeneous'.

Try solving it in your notebook first, then check the solution.

medium

Formulate a differential equation to model the temperature

T

of a body cooling in a room with a constant ambient temperature

T_a

, according to Newton's Law of Cooling, which states that the rate of change of temperature is proportional to the difference between the body's temperature and the ambient temperature. If a body cools from

80^\circ

C to

50^\circ

C in 20 minutes in a room at

20^\circ

C, create a formula for the temperature at time

t

.

Try solving it in your notebook first, then check the solution.

medium

Summarize the method of 'separation of variables' for solving a first-order, first-degree differential equation.

Try solving it in your notebook first, then check the solution.

medium

Propose a substitution to solve the differential equation

\frac{dy}{dx} = (4x+y+1)^2

. Justify your proposal by transforming the equation into a separable form, and then find the general solution.

Try solving it in your notebook first, then check the solution.

medium

Identify the order and degree of the differential equation

\left(\frac{d^3y}{dx^3}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0

. Explain your reasoning.

Try solving it in your notebook first, then check the solution.

medium

Identify if the differential equation

y' = \sin(x+y)

is homogeneous. Explain your reasoning.

Try solving it in your notebook first, then check the solution.

medium

Design a differential equation whose general solution represents all non-vertical straight lines passing through the point

(1, 2)

. Solve the created equation to verify your answer.

Try solving it in your notebook first, then check the solution.

medium

Find the general solution of the differential equation

(e^x + 1)y dy = (y+1)e^x dx

.

Try solving it in your notebook first, then check the solution.

hard

Justify why

e^{\int P dx}

is chosen as the integrating factor for the linear differential equation

\frac{dy}{dx} + Py = Q

. Derive this factor from the requirement that the left side of the equation becomes an exact derivative.

Try solving it in your notebook first, then check the solution.

hard

A tank contains 500 liters of pure water. Brine that contains 0.1 kg of salt per liter is pumped into the tank at a rate of 5 L/min. The well-mixed solution is pumped out at a rate of 10 L/min. Formulate a differential equation for the amount of salt

A(t)

in the tank at any time

t

. Propose a method to solve this equation and determine the amount of salt in the tank at the moment it becomes empty.

Try solving it in your notebook first, then check the solution.

hard

Solve the differential equation

(x+2y^2)\frac{dy}{dx} = y

and find the particular solution given that

y=1

when

x=1

.

Try solving it in your notebook first, then check the solution.

hard

Formulate the differential equation for the family of parabolas having their vertex at the origin and axis along the positive y-axis.

Try solving it in your notebook first, then check the solution.

hard

Create a differential equation for the family of curves where, for any point

(x,y)

on the curve, the length of the normal segment between the point and the x-axis is constant and equal to

k

. Solve this differential equation to identify the family of curves and justify your method. (Hint: Length of Normal =

y\sqrt{1 + (y')^2}

)

Try solving it in your notebook first, then check the solution.

hard

Solve the differential equation

x^2 dy + (xy+y^2)dx = 0

.

Try solving it in your notebook first, then check the solution.

hard

Propose a method to solve the Bernoulli differential equation

\frac{dy}{dx} + \frac{1}{x}y = x y^2

. Justify your proposed substitution by demonstrating that it transforms the equation into a linear differential equation. Then, find the general solution.

Try solving it in your notebook first, then check the solution.

hard

Critique the following reasoning: "To solve

(x+y)dx - (x-y)dy = 0

, we can rearrange it to

\frac{dy}{dx} = \frac{x+y}{x-y}

. Since this is a homogeneous equation, we substitute

y=vx

. This leads to

v + x \frac{dv}{dx} = \frac{x+vx}{x-vx} = \frac{1+v}{1-v}

. Therefore, the solution only depends on the ratio

y/x

." Is the conclusion fully justified? Propose a scenario where a particular solution might not be representable in this form.

Try solving it in your notebook first, then check the solution.

hard

Summarize the key characteristics of the three main types of first-order, first-degree differential equations: variable separable, homogeneous, and linear. For each type, describe its standard form and the initial step to solve it.

Try solving it in your notebook first, then check the solution.

hard

The rate of growth of a bacteria culture is proportional to the number present. If the count was 400 initially and 1000 after 1 hour, what will be the bacteria count after 1.5 hours?

Try solving it in your notebook first, then check the solution.

hard

Find the general solution for the differential equation:

y dx - (x+2y^2)dy = 0

.

Try solving it in your notebook first, then check the solution.

hard

Explain why the degree of the differential equation

\frac{d^2y}{dx^2} + \cos\left(\frac{dy}{dx}\right) = 0

is not defined.

Try solving it in your notebook first, then check the solution.

hard

Formulate a differential equation representing the family of parabolas with their vertex at the origin and axis along the positive y-axis.

Try solving it in your notebook first, then check the solution.

Differential EquationsClass 12 Mathematics Practice Questions