Quiz Tweek

Preparing your workspace...

Differential Equations Set-1

Practice Important MCQs for Differential Equations — Mathematics, Class 12.

Differential equations are essential in Class 12 Mathematics and in competitive exams because they model how quantities change with respect to other quantities (growth/decay, oscillations, electrical/physical systems). Mastery of solution techniques—like separable equations, linear/ Bernoulli methods, integrating factors, and Euler–Cauchy forms—directly improves both conceptual accuracy and speed in scoring high on board and competitive problems.

Minutes

Questions

1 / -0

Marking

Question Bank Preview

Q1. Solve the initial value problem $\dfrac{dy}{dx}=3x^{2}(1+y^{2}),\; y(0)=0$ .

(A)

$y=\arctan\!\bigl(x^{3}\bigr)$

(B)

$y=\tanh\!\bigl(x^{3}\bigr)$

(C)

$y=\tan\!\bigl(x^{3}\bigr)$

(D)

$y=-\tan\!\bigl(x^{3}\bigr)$

Q2. The one-parameter family $y=C e^{x}+\dfrac{1}{C}e^{-x}$ , $C\neq 0$ , is given. By eliminating $C$ obtain a first-order differential equation relating $y$ and $\dfrac{dy}{dx}$ .

(A)

$y^{2}-\displaystyle\left(\dfrac{dy}{dx}\right)^{2}=4$

(B)

$\displaystyle\left(\dfrac{dy}{dx}\right)^{2}-y^{2}=4$

(C)

$\displaystyle\dfrac{d^{2}y}{dx^{2}}-y=0$

(D)

$y^{2}+\displaystyle\left(\dfrac{dy}{dx}\right)^{2}=4$

Q3. Solve the initial value problem $\dfrac{dy}{dx}+\dfrac{1}{x}y=x^{2}y^{2},\; y(1)=1$ . (Assume $x>0$ .)

(A)

$y=\dfrac{2}{x(3+x^{2})}$

(B)

$y=\dfrac{2}{x(3-x^{2})}$

(C)

$y=\dfrac{2}{x^{2}(3-x^{2})}$

(D)

$y=\dfrac{2}{x(x^{2}-3)}$

Q4. Consider the differential form $M(x,y)\,dx+N(x,y)\,dy=0$ with $M,N$ having continuous partial derivatives in a region.
Assertion (A): If $\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}$ depends only on $x$ , then there exists an integrating factor $\mu(x)$ depending only on $x$ .
Reason (R): A necessary condition for an integrating factor $\mu(x)$ depending only on $x$ is that $\dfrac{\dfrac{\partial M}{\partial y}-\dfrac{\partial N}{\partial x}}{N}$ be a function of $x$ alone.

(A)

Both A and R are true and R is a correct explanation of A.

(B)

Both A and R are true but R is NOT a correct explanation of A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q5. Solve the initial value problem $\dfrac{d^{2}y}{dx^{2}}-4\dfrac{dy}{dx}+4y=e^{2x},\; y(0)=0,\; y'(0)=1$ .

(A)

$y=e^{2x}\!\left(x-\dfrac{x^{2}}{2}\right)$

(B)

$y=e^{2x}\!\left(x+x^{2}\right)$

(C)

$y=e^{2x}\!\left(x+\dfrac{x^{2}}{2}\right)$

(D)

$y=e^{2x}\!\left(1+x+\dfrac{x^{2}}{2}\right)$

Q6. Find the general solution of the differential equation $\dfrac{dy}{dx} + y = e^{x}\cos x$ .

(A)

$y=\dfrac{e^x(\cos x+\sin x)}{2}+C\,e^{-x}$

(B)

$y=\dfrac{e^x(2\cos x+\sin x)}{5}+C\,e^{-x}$

(C)

$y=\dfrac{e^x(\cos x-\sin x)}{5}+C\,e^{-x}$

(D)

$y=\dfrac{e^{-x}(2\cos x+\sin x)}{5}+C\,e^{x}$

Q7. Solve the initial value problem $\dfrac{dy}{dx} + \dfrac{2}{x}y = x^2 y^3,\quad y(1)=1$ . The solution for $x>0$ is

(A)

$y=\dfrac{1}{x^{3/2}\sqrt{2+x}}$

(B)

$y=\dfrac{1}{x^{3/2}\sqrt{2-x^{2}}}$

(C)

$y=-\dfrac{1}{x^{3/2}\sqrt{2-x}}$

(D)

$y=\dfrac{1}{x^{3/2}\sqrt{2-x}}$

Q8. Find the general solution of the Euler–Cauchy equation $x^2 y'' - 3x y' + 4y = 0$ for $x>0$ .

(A)

$y=C_1 x^2 + C_2 x^2\ln x$

(B)

$y=C_1 x + C_2 x^2$

(C)

$y=C_1 x^2 + C_2 x^{-2}$

(D)

$y=C_1 x^{1+i} + C_2 x^{1-i}$

Q9. For the differential equation $y = x\,\dfrac{dy}{dx} + \left(\dfrac{dy}{dx}\right)^2$ , the singular solution is

(A)

$y=Cx+C^2$

(B)

$y=\dfrac{x^2}{4}$

(C)

$y=-\dfrac{x^2}{4}$

(D)

$y=-\dfrac{1}{4}(x+C)^2$

Q10. Find the general solution of the ODE $y'' - 2y' + 5y = e^{x}\cos 2x$ .

(A)

$y=e^{x}(C_1\cos 2x + C_2\sin 2x) + \dfrac{x e^{x}}{2}\sin 2x$

(B)

$y=e^{x}(C_1\cos 2x + C_2\sin 2x) + \dfrac{x e^{x}}{4}\sin 2x$

(C)

$y=e^{x}(C_1\cos 2x + C_2\sin 2x) + \dfrac{x e^{x}}{4}\cos 2x$

(D)

$y=e^{x}(C_1\cos 2x + C_2\sin 2x) + \dfrac{e^{x}}{4}\sin 2x$

More Quizzes for Class 12

Explore other chapters and subjects, or view all quizzes for this class.

Differential Equations Set-1 MCQ Online Practice Test | Class 12 | Quiz Tweek

Quiz Tweek

Differential Equations Set-1

Question Bank Preview

More Quizzes for Class 12

Chemistry

Mathematics

Physics