Differential Equations: Class 12 Important Questions with Answers

Practice set: test yourself

Try every question before looking at the answers below. 64 or more out of 80 means you are ready for the board.

C. Very short answer (1 mark each)

1. Order and degree of $(y^{\prime }{)}^3+y=x$.
2. Order and degree of $y^{\prime \prime }+(y^{\prime }{)}^4=e^x$.
3. IF of $\frac{dy}{dx}+\frac{2y}{x}=x$.
4. IF of $\frac{dy}{dx}+y\tan x=\sec x$.
5. Solve $\frac{dy}{dx}=e^{x-y}$.
6. Solve $\frac{dy}{dx}=\frac{x}{y}$.
7. Solve $\frac{dy}{dx}={\sec }^{2}x$.
8. Is $x^2+xy$ homogeneous? Of what degree?
9. Solve $\frac{dy}{dx}=1+y^2$.
10. IF of $x\frac{dy}{dx}+y=x^3$.

D. Short answer I (2 marks each)

1. Solve $\frac{dy}{dx}=\frac{1+y^2}{1+x^2}$.
2. Solve $\frac{dy}{dx}=e^x{y}^2$.
3. Find the particular solution of $\frac{dy}{dx}=2xy$, $y(0)=3$.
4. Solve $x\frac{dy}{dx}=y+x$.
5. Solve $\frac{dy}{dx}+y=e^{-x}$.
6. Verify that $y=A\cos x+B\sin x$ is a solution of $y^{\prime \prime }+y=0$.
7. Solve $\frac{dy}{dx}=y\cot x$.
8. Solve $(1+x)dy=(1+y)dx$.

Rough work

E. Short answer II (3 marks each)

1. Solve $2xy\frac{dy}{dx}=x^2+y^2$.
2. Solve $\frac{dy}{dx}+2y=4x$.
3. Find the particular solution of $(1+x^2)\frac{dy}{dx}=1+y^2$, given $y(0)=1$.
4. Solve $x\frac{dy}{dx}-y=x^2\cos x$.
5. Solve $\frac{dx}{dy}+x=e^{-y}$.
6. A population grows at a rate proportional to its size and doubles in 5 years. In how many years will it triple? ($\log 2=0.6931$, $\log 3=1.0986$)

F. Long answer (5 marks each)

1. Solve the differential equation $(x-y)\frac{dy}{dx}=x+2y$.
2. Find the particular solution of $\frac{dy}{dx}+y\cot x=2\cos x$, given $y=0$ when $x=\frac{\pi }{2}$.

G. HOTS (higher-order thinking)

1. Why is the degree of $y^{\prime \prime }+\sin y^{\prime }=0$ not defined although its order is 2?
2. Why must $F$ have degree 0 for $y=vx$ to work?
3. Why does multiplying by the IF make the left side easy to integrate?
4. Can a DE be both variables separable and linear? Give an example.

20 MCQs

1. Order of $\frac{d^{3}y}{d{x}^3}+{\left(\frac{dy}{dx}\right)}^2=0$
   (a) 2 (b) 1 (c) 3 (d) 0
2. Degree of $(y^{\prime \prime }{)}^3+(y^{\prime }{)}^2=0$
   (a) 3 (b) 2 (c) 1 (d) not defined
3. Degree of $y^{\prime \prime }+e^{y^{\prime }}=0$
   (a) 1 (b) 0 (c) 2 (d) not defined
4. Arbitrary constants in the general solution of a 2nd-order DE
   (a) 1 (b) 2 (c) 0 (d) 3
5. IF of $\frac{dy}{dx}+3y=x$
   (a) $e^{-3x}$ (b) $e^{3x}$ (c) $3x$ (d) $x^3$
6. IF of $\frac{dy}{dx}-y=e^x$
   (a) $e^x$ (b) $x$ (c) $-x$ (d) $e^{-x}$
7. IF of $x\frac{dy}{dx}+2y=x^2$
   (a) $x^2$ (b) $2x$ (c) $e^{x^2}$ (d) $\frac{1}{x^2}$
8. Solution of $\frac{dy}{dx}=x^2$
   (a) $y=2x+C$ (b) $y=x^3+C$ (c) $y=\frac{x^3}{3}+C$ (d) $y=3x^2+C$
9. Solution of $\frac{dy}{dx}=-\frac{x}{y}$
   (a) $x^2-y^2=C$ (b) $y=Cx$ (c) $xy=C$ (d) $x^2+y^2=C$
10. Solution of $\frac{dy}{dx}=\frac{y}{x}$
    (a) $xy=C$ (b) $y=Cx$ (c) $y=x+C$ (d) $y=C{x}^2$
11. Substitution for $\frac{dx}{dy}=h\left(\frac{x}{y}\right)$
    (a) $y=vx$ (b) $xy=v$ (c) $x=vy$ (d) $x=v$
12. Which is homogeneous?
    (a) $y^{\prime }=\frac{x^2+y^2}{xy}$ (b) $y^{\prime }=x+y+1$ (c) $y^{\prime }=x^2+y$ (d) $y^{\prime }=\sin x+y$
13. Which is linear?
    (a) $y^{\prime }+y^2=x$ (b) $y^{\prime }+xy=x^2$ (c) $y{y}^{\prime }+y=x$ (d) $(y^{\prime }{)}^2+y=0$
14. Solution of $\frac{dy}{dx}=2y$
    (a) $y=2x+C$ (b) $y=e^x+C$ (c) $y=C{e}^{2x}$ (d) $y=C{x}^2$
15. Particular solution of $\frac{dy}{dx}=2x$, $y(0)=1$
    (a) $y=x^2$ (b) $y=x^2-1$ (c) $y=2x+1$ (d) $y=x^2+1$
16. IF of $\frac{dx}{dy}+\frac{x}{y}=y$
    (a) $y$ (b) $x$ (c) $\frac{1}{y}$ (d) $e^y$
17. Order + degree of $y^{\prime \prime }+3(y^{\prime }{)}^3=0$
    (a) 3 (b) 5 (c) 4 (d) 2
18. Solution of $\frac{dy}{dx}=e^{-y}$
    (a) $e^{-y}=x+C$ (b) $y=-e^{-x}+C$ (c) $y=e^x+C$ (d) $e^y=x+C$
19. $P=P_0{e}^{kt}$ is the solution of
    (a) $\frac{dP}{dt}=k$ (b) $\frac{dP}{dt}=kP$ (c) $\frac{dP}{dt}=kt$ (d) $\frac{dP}{dt}=\frac{P}{k}$
20. IF of $(1+x^2)\frac{dy}{dx}+2xy=1$
    (a) $e^{x^2}$ (b) $2x$ (c) $1+x^2$ (d) $\log (1+x^2)$

1 (c) · 2 (a) · 3 (d) · 4 (b) · 5 (b) · 6 (d) · 7 (a) · 8 (c) · 9 (d) · 10 (b) · 11 (c) · 12 (a) · 13 (b) · 14 (c) · 15 (d) · 16 (a) · 17 (a) · 18 (d) · 19 (b) · 20 (c)

Assertion–Reason (6)

(a) Both true, R explains A. (b) Both true, R does not explain A. (c) A true, R false. (d) A false, R true.

1. A: The degree of $(y^{\prime \prime \prime }{)}^2+y^{\prime \prime }=0$ is 2. R: The degree is the power of the highest-order derivative when the equation is a polynomial in the derivatives.
2. A: $\frac{dy}{dx}=\frac{x+y}{x-y}$ is homogeneous. R: $F(\lambda x,\lambda y)=F(x,y)$ for $F=\frac{x+y}{x-y}$.
3. A: The IF of $\frac{dy}{dx}+\frac{y}{x}=x^3$ is $x$. R: The IF of $y^{\prime }+Py=Q$ is $e^{\int Pdx}$.
4. A: The general solution of a second-order DE has two arbitrary constants. R: Every second-order DE is linear.
5. A: $\frac{dy}{dx}+y^2=x$ is a linear DE. R: In a linear DE, $y$ and $\frac{dy}{dx}$ occur only in the first degree.
6. A: $y=C{e}^x$ is the general solution of $\frac{dy}{dx}=y$. R: The degree of $\frac{dy}{dx}=y$ is 1.

Key: 1 (a) · 2 (a) · 3 (a) · 4 (c) · 5 (d) · 6 (b)

Answers to the practice set