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Differential Equations Set-4 MCQ Online Practice Test | Class 12 | Quiz Tweek

Differential Equations Set-4

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

Differential Equations is a core Class 12 chapter that builds your ability to model growth/decay, mechanics-like rate laws, and obtain solutions using standard techniques. On boards and competitive exams, questions from this chapter directly test your command over separation of variables, linear/Bernoulli equations, Clairaut equations, orthogonal trajectories, Euler–Cauchy methods, and envelope/singular solution concepts—skills that frequently appear in both theory-linked and purely calculation-based MCQs.

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

(A)

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

(B)

( $y^{2}=x^{3}+4$ )

(C)

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

(D)

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

Q2. Solve the Bernoulli equation $\dfrac{dy}{dx}+\dfrac{1}{x}y=x^{2}y^{2}$ with $y(1)=\dfrac{1}{2}$ . The particular solution is:

(A)

$\displaystyle\left(\frac{2}{x(5-x^{2})}\right)$

(B)

$\displaystyle\left(\frac{2}{5-x^{2}}\right)$

(C)

$\displaystyle\left(\frac{2}{x(5+x^{2})}\right)$

(D)

$\displaystyle\left(\frac{2x}{5-x^{2}}\right)$

Q3. Find the general solution of the differential equation $\dfrac{dy}{dx}=\dfrac{x^{2}+y^{2}}{2xy}$ .

(A)

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

(B)

( $x^{2}+y^{2}=Cx$ )

(C)

( $x^{2}-y^{2}=Cx^{2}$ )

(D)

( $x^{2}-y^{2}=Cx$ )

Q4. Assertion (A): The initial value problem $\dfrac{dy}{dx}=y^{1/3},\; y(0)=0$ has infinitely many solutions.
Reason (R): The function $f(y)=y^{1/3}$ is not Lipschitz continuous in any neighbourhood of $y=0$ .

(A)

(Both A and R are true and R is the correct explanation of A.)

(B)

(Both A and R are true but R is not the correct explanation of A.)

(C)

(A is true but R is false.)

(D)

(A is false and R is true.)

Q5. Consider the differential equation $y = x\,\dfrac{dy}{dx} +\left(\dfrac{dy}{dx}\right)^{2}$ . Which of the following gives the general solution and the singular solution?

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

(D)

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

Q6. Solve the initial value problem $\dfrac{dy}{dx}+2y=e^{-x}$ with $y(0)=1$ . The particular solution is

(A)

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

(B)

$y=e^{-x}$

(C)

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

(D)

$y=e^{x}$

Q7. Find the particular solution of $\dfrac{dy}{dx}+\dfrac{2}{x}y=x^{2}y^{3}$ satisfying $y(1)=1$ for $x>0$ .

(A)

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

(B)

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

(C)

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

(D)

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

Q8. The family of curves is given by $y^{2}=Cx^{3}$ (parameter $C$ ). The orthogonal trajectories of this family are

(A)

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

(B)

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

(C)

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

(D)

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

Q9. Solve the differential equation $(x^{2}+y^{2})\,dx-2xy\,dy=0$ . Its general solution can be written as

(A)

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

(B)

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

(C)

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

(D)

$x^{2}+y^{2}=Cx$

Q10. Consider the Clairaut equation $y=x\dfrac{dy}{dx}+\left(\dfrac{dy}{dx}\right)^{3}$ . The singular (envelope) solution is

(A)

$y=Cx+C^{3}$

(B)

$27y^{2}+4x^{3}=0$

(C)

$y^{2}=\dfrac{4}{27}x^{3}$

(D)

$y^{2}=\dfrac{4}{27}x^{2}$

Q11. Solve the differential equation

\frac{dy}{dx}=\frac{y-x}{x},\quad x>0

with $y(1)=2$ .

(A)

$y=-x\ln x + x$

(B)

$y=-x\ln x + 2x$

(C)

$y= x\ln x + 2x$

(D)

$y=-\ln x + 2x$

Q12. Find the general solution of the differential equation

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

(A)

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

(B)

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

(C)

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

(D)

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

Q13. Find the general solution of the Euler–Cauchy equation

x^2y''-3xy'+4y=0,\quad x>0.

(A)

$y=x^2\bigl(C_1+C_2\ln x\bigr)$

(B)

$y=C_1x^2+C_2x$

(C)

$y=C_1x^2+C_2x^{-2}$

(D)

$y=C_1x^{1+\sqrt{3}}+C_2x^{1-\sqrt{3}}$

Q14. For the differential equation

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

determine the general solution and any singular solution(s).

(A)

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

(B)

General: $y=Cx-C^2$ , Singular: none

(C)

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

(D)

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

Q15. Consider the differential equation

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

Which of the following statements is true?

(A)

The general solution is $\dfrac{y}{x^2-y^2}=C$ and the lines $y=\pm x$ are contained in this family.

(B)

The general solution is $\dfrac{y}{x^2-y^2}=C$ , while the lines $y=\pm x$ are singular (envelope) solutions not contained in the general family.

(C)

The general solution is $\dfrac{y}{x^2+y^2}=C$ and there are no singular solutions.

(D)

The general solution is $\ln\!\left|\dfrac{y}{x^2-y^2}\right|=\ln|x|+C$ and the lines $y=\pm x$ arise from this family for $C=0$ .

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