Quiz Tweek

Preparing your workspace...

Differential Equations Set-6 MCQ Online Practice Test | Class 12 | Quiz Tweek

Differential Equations Set-6

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

Differential Equations are central to Class 12 Mathematics because they model changing systems (growth/decay, oscillations, and flows) and build core techniques like linear ODEs, Bernoulli substitutions, and orthogonal trajectories. These methods are frequently used in both board exams and competitive tests to assess concept clarity and algebraic accuracy under different forms.

Minutes

Questions

1 / -0

Marking

Question Bank Preview

Q1. Solve the differential equation

\frac{dy}{dx}+\frac{2}{x}y=x^2,\qquad x>0.

The general solution is:

(A)

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

(B)

$y=\dfrac{x^3}{5}+Cx^2$

(C)

$y=\dfrac{x^3}{5}+\dfrac{C}{x^2}$

(D)

$y=\dfrac{3x^3}{5}+\dfrac{C}{x}$

Q2. Solve the Bernoulli equation

\frac{dy}{dx}+\frac{1}{x}y=x^2y^3,\qquad x>0.

The general solution is:

(A)

$y=\pm\displaystyle\frac{1}{x\sqrt{C+2x}}$

(B)

$y=\pm\displaystyle\frac{1}{x\sqrt{C-2x}}$

(C)

$y=\displaystyle\frac{1}{x\sqrt{2x-C}}$

(D)

$y=\pm\displaystyle\frac{x}{\sqrt{C-2x}}$

Q3. Find the particular solution of

\frac{dy}{dx}=\frac{x+y+1}{x-y+1}

that satisfies $y(0)=0$ . An implicit form of the solution is:

(A)

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

(B)

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

(C)

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

(D)

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

Q4. Consider the differential equation of Clairaut type

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

The general solution and any singular solution are:

(A)

General: $y=Cx+C^2$ ;; Singular: $y=-\dfrac{x^2}{4}$

(B)

General: $y=Cx-C^2$ ;; Singular: $y=\dfrac{x^2}{4}$

(C)

General: $y=C+C^2x$ ;; Singular: none

(D)

General: $y=Cx+C^2$ ;; Singular: $y=\dfrac{x^2}{4}$

Q5. Solve the second-order differential equation

y\,y''-(y')^2=0.

The general solution is:

(A)

$y=C_1+C_2x$

(B)

$y=C_1e^{C_2x}$

(C)

$y=C_1x^{C_2}$

(D)

$y=C_1e^{C_2x^2}$

Q6. Solve the initial value problem

\frac{dy}{dx}=\frac{3x^2}{2y},\qquad y(1)=1.

The value of $y(2)$ is:

(A)

$-2\sqrt{2}$

(B)

$4$

(C)

$2\sqrt{2}$

(D)

$8$

Q7. Find the general solution of the Bernoulli equation

\frac{dy}{dx}+\frac{2}{x}y=x^3y^2,\qquad x\ne 0.

The general solution is:

(A)

$y=\dfrac{1}{x^2\left(C-\dfrac{x^2}{2}\right)}$

(B)

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

(C)

$y=\dfrac{1}{Cx^2-\dfrac{x^4}{4}}$

(D)

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

Q8. Find the general solution of

y''-2y'+5y=e^{x}\cos 2x.

(A)

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

(B)

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

(C)

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

(D)

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

Q9. Consider the initial value problem

\frac{dy}{dx}=3y^{2/3},\qquad y(0)=0.

Which one of the following statements about existence and uniqueness of solutions on $\mathbb{R}$ is correct?

(A)

The IVP has exactly two solutions, $y\equiv 0$ and $y=x^3$ .

(B)

There are infinitely many solutions: for any $a\ge 0$ the function defined by $y(x)=0$ for $x\le a$ and $y(x)=(x-a)^3$ for $x>a$ , together with $y\equiv 0$ , all satisfy the IVP.

(C)

The IVP has a unique solution $y(x)=x^3$ .

(D)

No solution exists on the whole real line.

Q10. Solve the differential equation

\frac{dy}{dx}=\frac{x^2+y^2}{2xy},\qquad x\ne 0,\;y\ne 0.

The general implicit solution can be written as:

(A)

$y^2=x^2-Cx,\quad C\in\mathbb{R}$

(B)

$y^2=x^2+Cx,\quad C\in\mathbb{R}$

(C)

$\ln\!\left|1-\dfrac{y^2}{x^2}\right|=\ln|x|+C$

(D)

$y=x\sqrt{1+\dfrac{C}{x}}$

Q11. Find the general solution of the differential equation $\dfrac{dy}{dx} + y\tan x = \sin x$ .

(A)

$y = \cos x\bigl(\ln|\cos x| + C\bigr)$

(B)

$y = \cos x\bigl(C - \ln|\cos x|\bigr)$

(C)

$y = \sec x\bigl(-\ln|\cos x| + C\bigr)$

(D)

$y = \sin x\bigl(-\ln|\cos x| + C\bigr)$

Q12. Solve the Bernoulli equation $\dfrac{dy}{dx} + \dfrac{2}{x}y = x^3 y^2,\; x>0$ . What is the general solution?

(A)

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

(B)

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

(C)

$y=\dfrac{1}{C x^2 - \dfrac{x^2}{2}}$

(D)

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

Q13. The family of circles $x^2+y^2=C\,y$ is given. The family of curves orthogonal to these circles is:

(A)

$x^2+y^2=K\,x$

(B)

$x^2+y^2=K\,y$

(C)

$x^2+y^2=K$

(D)

$x^2-y^2=K\,x$

Q14. Consider the initial value problem $\dfrac{dy}{dx}=y^{1/3},\; y(0)=0$ . Which of the following is true about solutions through $(0,0)$ ?

(A)

There is exactly one solution satisfying the IVP.

(B)

There are exactly two distinct solutions satisfying the IVP.

(C)

Infinitely many solutions exist; besides $y\equiv0$ there are solutions that stay $0$ up to some $x=a\ge0$ and then follow $y(x)=\bigl(\tfrac{2}{3}(x-a)\bigr)^{3/2}$ for $x\ge a$ .

(D)

No solution exists through $(0,0)$ .

Q15. Find the general (implicit) solution of the differential equation $\dfrac{dy}{dx}=\dfrac{x+2y}{2x-y}$ .

(A)

$2\arctan\!\bigl(\tfrac{y}{x}\bigr)-\tfrac{1}{2}\ln\!\bigl(x^{2}+y^{2}\bigr)=C$

(B)

$4\arctan\!\bigl(\tfrac{y}{x}\bigr)-\tfrac{1}{2}\ln\!\bigl(x^{2}+y^{2}\bigr)=C$

(C)

$2\arctan\!\bigl(\tfrac{y}{x}\bigr)+\ln\!\bigl(x^{2}+y^{2}\bigr)=C$

(D)

$4\arctan\!\bigl(\tfrac{y}{x}\bigr)+\ln\!\bigl(x^{2}+y^{2}\bigr)=C$

...and 5 more challenging questions available in the interactive simulator.

More Quizzes for Class 12

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

Quiz Tweek

Differential Equations Set-6

Question Bank Preview

More Quizzes for Class 12

Chemistry

Mathematics

Physics