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Differential Equations Set-5 MCQ Online Practice Test | Class 12 | Quiz Tweek

Differential Equations Set-5

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

Differential equations are central to Class 12 Mathematics and frequently appear in board and competitive exams because they model real-world change (growth, decay, motion, heat, circuits) and train you to apply key solution techniques like linearization, integrating factors, Bernoulli transformations, separable/homogeneous forms, and Clairaut-type envelopes. Mastery of this chapter directly improves speed and accuracy in solving both standard and tricky match-the-following problems.

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Q1. Solve the initial value problem

\frac{dy}{dx}+2y=3e^{2x},\qquad y(0)=1

and compute $y(\ln 2)$ .

(A)

$\dfrac{48}{16}$

(B)

$\dfrac{44}{16}$

(C)

$\dfrac{49}{16}$

(D)

$\dfrac{52}{16}$

Q2. Solve the differential equation

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

The general solution can be written as:

(A)

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

(B)

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

(C)

$y=\pm\dfrac{1}{x^{2}\sqrt{2+Cx}}$

(D)

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

Q3. Find the general solution of the differential equation

\frac{d^2y}{dx^2}=\frac{\left(\dfrac{dy}{dx}\right)^2}{y},\qquad y>0.

(A)

$y=C_1+C_2e^{x}$

(B)

$y=C_1x+C_2$

(C)

$y=\dfrac{1}{C_1x+C_2}$

(D)

$y=C_1e^{C_2x}$

Q4. Consider the Clairaut equation

y=x\frac{dy}{dx}+e^{\frac{dy}{dx}}.

The singular (envelope) solution is:

(A)

$y=x\ln(-x)-x$

(B)

$y=cx+e^{c}\quad(\text{family of straight lines})$

(C)

$y=x\ln x-x$

(D)

$y=-x\ln(-x)-x$

Q5. Solve the differential equation

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

and express the general solution implicitly.

(A)

$x^2+y^2=Cx$

(B)

$x^2-y^2=Cx$

(C)

$x^2-y^2=C$

(D)

$y^2-x^2=Cx$

Q6. Solve the initial value problem $\dfrac{dy}{dx} + 2y = e^{-x},\; y(0)=\tfrac{1}{2}$ . Which of the following is the solution?

(A)

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

(B)

$y=e^{-x}-e^{-2x}$

(C)

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

(D)

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

Q7. Solve the initial value problem $x\dfrac{dy}{dx}+2y=x^{4}y^{2},\; y(1)=1$ . For $x$ in the interval where the solution is defined, $y(x)$ equals:

(A)

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

(B)

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

(C)

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

(D)

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

Q8. Consider the differential equation $y = x\frac{dy}{dx} + \dfrac{1}{\frac{dy}{dx}}\,$ for $x>0$ . The general solution is a one-parameter family of straight lines. Which of the following represents the singular solution (envelope) of that family?

(A)

$y = c\,x + \dfrac{1}{c}$

(B)

$y = 2\sqrt{x}$

(C)

$y = -\,2\sqrt{x}$

(D)

$y^{2}=4x$

Q9. Solve the differential equation $\dfrac{dy}{dx}=\dfrac{x+y+1}{x-y+2}$ and choose the correct implicit general solution (constant $C$ arbitrary). Note the substitution that reduces it to a homogeneous form is $X=x+\tfrac{3}{2},\;Y=y-\tfrac{1}{2}$ .

(A)

$\displaystyle \arctan\!\left(\frac{y-\tfrac{1}{2}}{\,x+\tfrac{3}{2}\,}\right)+\tfrac{1}{2}\ln\!\left(1+\left(\frac{y-\tfrac{1}{2}}{\,x+\tfrac{3}{2}\,}\right)^{2}\right)=\ln\!\big|x+\tfrac{3}{2}\big|+C$

(B)

$\displaystyle \arctan\!\left(\frac{y-\tfrac{1}{2}}{\,x+\tfrac{3}{2}\,}\right)-\tfrac{1}{2}\ln\!\left(1+\left(\frac{y-\tfrac{1}{2}}{\,x+\tfrac{3}{2}\,}\right)^{2}\right)=\ln\!\big|x+\tfrac{3}{2}\big|+C$

(C)

(D)

$\displaystyle \arctan\!\left(\frac{y-\tfrac{1}{2}}{\,x+\tfrac{3}{2}\,}\right)-\ln\!\left(1+\left(\frac{y-\tfrac{1}{2}}{\,x+\tfrac{3}{2}\,}\right)^{2}\right)=\ln\!\big|x+\tfrac{3}{2}\big|+C$

Q10. The family of circles through $(0,0)$ and $(1,0)$ is given by $x^{2}+y^{2}-x+by=0$ (parameter $b$ ). Find the orthogonal trajectories of this family (general solution with arbitrary constant $C$ ).

(A)

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

(B)

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

(C)

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

(D)

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

Q11. Solve the initial value problem $\dfrac{dy}{dx} = \dfrac{3x^2}{1+y^2},\; y(0)=0$ . Which of the following represents the solution?

(A)

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

(B)

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

(C)

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

(D)

$y^{2}=x^{3}$

Q12. Solve the initial value problem $\dfrac{dy}{dx}+\dfrac{2}{x}y=x^{2}y^{3},\; y(1)=1$ . Which of the following is the solution valid near $x=1$ ?

(A)

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

(B)

$y=\dfrac{1}{\sqrt{2x^{3}+x^{4}}}$

(C)

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

(D)

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

Q13. Solve the initial value problem $\dfrac{dy}{dx}=\dfrac{x+y}{x-y},\; y(1)=1$ . The implicit solution can be written in terms of $v=\dfrac{y}{x}$ . Which of the following implicit relations (with the constant evaluated using $y(1)=1$ ) is correct?

(A)

$\displaystyle \arctan\!\left(\dfrac{y}{x}\right)-\dfrac{1}{2}\ln\!\left(1+\left(\dfrac{y}{x}\right)^{2}\right)=\ln|x|+\dfrac{\pi}{4}+\dfrac{1}{2}\ln 2$

(B)

$\displaystyle \arctan\!\left(\dfrac{y}{x}\right)-\dfrac{1}{2}\ln\!\left(1+\left(\dfrac{y}{x}\right)^{2}\right)=\ln|x|-\dfrac{\pi}{4}+\dfrac{1}{2}\ln 2$

(C)

$\displaystyle \arctan\!\left(\dfrac{y}{x}\right)+\dfrac{1}{2}\ln\!\left(1+\left(\dfrac{y}{x}\right)^{2}\right)=\ln|x|+\dfrac{\pi}{4}-\dfrac{1}{2}\ln 2$

(D)

$\displaystyle \arctan\!\left(\dfrac{y}{x}\right)-\dfrac{1}{2}\ln\!\left(1+\left(\dfrac{y}{x}\right)^{2}\right)=\ln|x|+\dfrac{\pi}{4}-\dfrac{1}{2}\ln 2$

Q14. Consider the one-parameter family of straight lines $y=cx+e^{c}$ , $c\in\mathbb{R}$ . The envelope (singular solution) of this family is obtained by eliminating $c$ . Which of the following is the envelope (state domain where necessary)?

(A)

$y=x\ln x + x$

(B)

$y=x\ln(-x)-x\quad (x<0)$

(C)

$y=-x\ln(-x)-x$

(D)

$y=-x\ln x + x$

Q15. Find the general solution (for $x>0$ ) of the Cauchy–Euler equation $x^{2}y''-3xy'+4y=x^{2}\ln x$ . Which of the following gives the general solution?

(A)

$y=x^{2}\!\left(C_{1}+C_{2}\ln x+\dfrac{1}{2}(\ln x)^{3}\right)$

(B)

$y=x^{2}\!\left(C_{1}+C_{2}\ln x+\dfrac{1}{6}(\ln x)^{2}\right)$

(C)

$y=x^{2}\!\left(C_{1}+C_{2}\ln x+\dfrac{1}{6}(\ln x)^{3}\right)$

(D)

$y=x^{2}\!\left(C_{1}+C_{2}\ln x\right)+\dfrac{1}{6}(\ln x)^{3}$

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

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