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Differential Equations Set-3 MCQ Online Practice Test | Class 12 | Quiz Tweek

Differential Equations Set-3

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

Differential Equations are fundamental in Class 12 Mathematics because they model change in physics, engineering, and biology. Mastery of basic types (linear, Bernoulli, Clairaut), methods (integrating factor, substitutions), and concept of singular solutions is essential for scoring well on board exams as well as competitive tests like JEE/NEET, where application-based reasoning from differential equations is frequently asked.

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

(A)

$y=\dfrac{1}{5}e^{3x}-\dfrac{4}{5}e^{-2x}$

(B)

$y=\dfrac{1}{5}e^{3x}+e^{-2x}$

(C)

$y=\dfrac{1}{5}e^{3x}+\dfrac{4}{5}e^{-2x}$

(D)

$y=\dfrac{1}{5}e^{-3x}+\dfrac{4}{5}e^{-2x}$

Q2. Find the general solution of the Bernoulli-type differential equation $\dfrac{dy}{dx}+y\tan x = y^{2}\sec x$ .

(A)

$y=\dfrac{\cos x}{C-x}$

(B)

$y=\dfrac{\sin x}{C-x}$

(C)

$y=\dfrac{\cos x}{C+x}$

(D)

$y=\dfrac{1}{\,C-\sin x\,}$

Q3. Solve the differential equation $(x y + x)\,dx + x^{2}\,dy = 0,\; x\ne 0$ . The general solution is

(A)

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

(B)

$y=Cx-1$

(C)

$y=\dfrac{x-C}{x}$

(D)

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

Q4. Consider the differential equation $y = x\dfrac{dy}{dx} +\left(\dfrac{dy}{dx}\right)^{2}$ .
Assertion (A): The singular solution is $y=-\dfrac{x^{2}}{4}$ .
Reason (R): For Clairaut's equation $y = x p + p^{2}$ (with $p=\dfrac{dy}{dx}$ ) the singular solution is the envelope of the family of straight lines $y = Cx + C^{2}$ .

(A)

Both A and R are true but R does not correctly explain A.

(B)

Both A and R are true and R correctly explains A.

(C)

A is true but R is false.

(D)

A is false but R is true.

Q5. For the initial value problem $\dfrac{dy}{dx}=y^{1/3},\; y(0)=0$ , which one of the following statements is correct?

(A)

The IVP has infinitely many solutions; besides $y\equiv 0$ , for any $a\ge 0$ the function $y(x)=0$ for $x\le a$ and $y(x)=\left(\dfrac{2}{3}(x-a)\right)^{3/2}$ for $x\ge a$ is a solution.

(B)

The IVP has a unique solution $y(x)\equiv 0$ for all real $x$ .

(C)

There is exactly one nonzero solution $y(x)=\left(\dfrac{2}{3}x\right)^{3/2}$ for $x\ge 0$ and $y(x)=0$ for $x<0$ .

(D)

The general solution is $y=\left(\dfrac{2}{3}(x+C)\right)^{3/2}$ valid for all real $x$ , hence uniqueness holds.

Q6. Find the particular solution of the differential equation

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

satisfying $y(1)=1$ .

(A)

$y=\dfrac{x^3}{5} + \dfrac{1}{5x^2}$

(B)

$y=\dfrac{x^5 + 4}{5x^2}$

(C)

$y=\dfrac{x^3}{5} - \dfrac{4}{5x^2}$

(D)

$y=\dfrac{x^3}{5} + \dfrac{4}{5}x^2$

Q7. Find the general solution of the differential equation

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

(A)

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

(B)

$y=\dfrac{2}{x\bigl(x^2 - 2C\bigr)}$

(C)

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

(D)

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

Q8. The one-parameter family of circles

x^2 + y^2 = 2ax\quad (a\ \text{parameter})

is given. The orthogonal trajectories of this family are described by

(A)

$x^2 + y^2 = C y$

(B)

$x^2 + y^2 = -C y$

(C)

$x^2 + y^2 = C x$

(D)

$y^2 - x^2 = C y$

Q9. Consider the differential equation

\bigl(y'\bigr)^2 + x y' - y = 0.

Its general integral is $y = Cx + C^2$ . The singular (envelope) solution of this differential equation is

(A)

No singular solution exists

(B)

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

(C)

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

(D)

$y=Cx + C^2$

Q10. Find the general solution of the differential equation

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

by using an integrating factor of the form $x^m y^n$ .

(A)

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

(B)

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

(C)

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

(D)

$-\dfrac{1}{x y} - \dfrac{1}{3x^3} = C$

Q11. Solve the initial value problem $\dfrac{dy}{dx} + \dfrac{2}{x}y = x^2,\; y(1)=0$ (for $x\neq 0$ ).

(A)

$y=\dfrac{x^3}{5}+\dfrac{1}{5x^2}$

(B)

$y=\dfrac{x^3}{3}-\dfrac{1}{5x^2}$

(C)

$y=\dfrac{x^3}{5}-\dfrac{1}{5x^2}$

(D)

$y=\dfrac{x^3}{5}-\dfrac{1}{x^2}$

Q12. Solve the initial value problem $\dfrac{dy}{dx} + \dfrac{1}{x}y = x^2 y^3,\; y(1)=1$ .

(A)

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

(B)

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

(C)

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

(D)

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

Q13. Find the general solution of the differential equation $\dfrac{dy}{dx}=\dfrac{x+y}{x-y}\,$ $(x\neq y)$ .

(A)

$\displaystyle \arctan\!\left(\frac{y}{x}\right)+\tfrac{1}{2}\ln\!\left(1+\left(\frac{y}{x}\right)^2\right)=\ln|x|+C$

(B)

$\displaystyle \arctan\!\left(\frac{x}{y}\right)-\tfrac{1}{2}\ln\!\left(1+\left(\frac{y}{x}\right)^2\right)=\ln|x|+C$

(C)

$\displaystyle \arctan\!\left(\frac{y}{x}\right)-\tfrac{1}{2}\ln|x|-\tfrac{1}{2}\ln\!\left(1+\left(\frac{y}{x}\right)^2\right)=C$

(D)

$\displaystyle \arctan\!\left(\frac{y}{x}\right)-\tfrac{1}{2}\ln\!\left(1+\left(\frac{y}{x}\right)^2\right)=\ln|x|+C$

Q14. For the Clairaut equation $y = x\,p + \dfrac{1}{p}$ where $p=\dfrac{dy}{dx}$ , determine the general solution and the singular solution(s).

(A)

General solution $y=Cx+\dfrac{1}{C}$ ; singular solution $y=2\sqrt{x}$

(B)

General solution $y=Cx+\dfrac{1}{C}$ ; singular solution $y^2=4x$

(C)

General solution $y=Cx-\dfrac{1}{C}$ ; singular solution $y^2=4x$

(D)

General solution $y=Cx+\dfrac{1}{C}$ ; singular solution $y=0$

Q15. Solve the initial value problem $y''-2y'+y=e^{x},\; y(0)=0,\; y'(0)=1$ .

(A)

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

(B)

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

(C)

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

(D)

$y=\dfrac{1}{2}x^2 e^{x}$

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