Vidaara.orgClass 12 · Mathematics
CodeVID-M12-09-CT
Differential Equations — Full Chapter Test
Name: ____________________
Roll No.: __________
Date: ____________
General Instructions
- This is a full-length test covering the whole chapter — every topic is included.
- All questions are compulsory.
- Section A carries 1 mark each, Section B 2 marks, Section C 3 marks and Section D 5 marks.
- Show all working for Sections B, C and D. Only final answers are given at the end — for full solutions, raise your doubts with your teacher.
Section A — Multiple Choice Questions
5 × 1 = 5 marks
1.
The degree of $\left(\dfrac{d^2y}{dx^2}\right)^3+\dfrac{dy}{dx}=0$ is:
- A.$1$
- B.$2$
- C.$3$
- D.undefined
2.
Solving $\dfrac{dy}{dx}=\dfrac{x}{y}$ gives:
- A.$y^2-x^2=C$
- B.$y=Cx$
- C.$x^2+y^2=C$
- D.$y=x+C$
3.
The integrating factor of $\dfrac{dy}{dx}+\dfrac{y}{x}=x$ is:
- A.$x$
- B.$\ln x$
- C.$e^{x}$
- D.$\tfrac1x$
4.
The order of $\dfrac{dy}{dx}+y=x$ is:
- A.$1$
- B.$2$
- C.$0$
- D.$3$
5.
The solution of $\dfrac{dy}{dx}=ky$ is:
- A.$y=kx+C$
- B.$y=Ce^{kx}$
- C.$y=Cx^k$
- D.$y=\ln(kx)$
Section B — Short Answer (2 marks)
4 × 2 = 8 marks
6.
Find the order and degree of $\dfrac{d^2y}{dx^2}+3\left(\dfrac{dy}{dx}\right)^2+y=0$.
7.
Solve $\dfrac{dy}{dx}=2x$.
8.
Find the IF of $\dfrac{dy}{dx}+3y=x$.
9.
Find the order and degree of $\left(\dfrac{dy}{dx}\right)^3+2y=x$.
Section C — Short Answer (3 marks)
4 × 3 = 12 marks
10.
Find the order and degree of $\left(\dfrac{d^2y}{dx^2}\right)^2+\left(\dfrac{dy}{dx}\right)^3+y=0$.
11.
Solve $\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}$.
12.
Solve $\dfrac{dy}{dx}+\dfrac{y}{x}=x$.
13.
Form the differential equation of the family $y=mx+c$.
Section D — Long Answer (5 marks)
2 × 5 = 10 marks
14.
Form the differential equation of the family of circles $x^2+y^2=a^2$.
15.
Solve $(1+x^2)\dfrac{dy}{dx}=1$, given $y(0)=0$.
Answer Key
Section A — Multiple Choice Questions
- (C) $3$
- (A) $y^2-x^2=C$
- (A) $x$
- (A) $1$
- (B) $y=Ce^{kx}$
Section B — Short Answer (2 marks)
- Order $2$, degree $1$.
- $y=x^2+C$.
- $e^{3x}$.
- Order $1$, degree $3$.
Section C — Short Answer (3 marks)
- Order $2$, degree $2$.
- $\tan^{-1}y=\tan^{-1}x+C$.
- $y=\tfrac{x^2}{3}+\tfrac{C}{x}$.
- $\dfrac{d^2y}{dx^2}=0$.
Section D — Long Answer (5 marks)
- $x+y\dfrac{dy}{dx}=0$.
- $y=\tan^{-1}x$.
Generated by Vidaara.org · Assignment VID-M12-09-CT · vidaara.org