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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-09-LIN-01
Linear Differential Equations — Assignment
Chapter: Differential Equations
Topic: Linear Differential Equations
Maximum Marks: 35
Time: 75 minutes
Name: ____________________ Roll No.: __________ Date: ____________

General Instructions

  • 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 integrating factor of $\dfrac{dy}{dx}+\dfrac{y}{x}=x$ is:
  • A.$x$
  • B.$\ln x$
  • C.$e^{x}$
  • D.$\tfrac1x$
2.
For $\dfrac{dy}{dx}+2y=e^x$, the IF is:
  • A.$e^{x}$
  • B.$e^{2x}$
  • C.$2x$
  • D.$e^{-2x}$
3.
A linear first-order DE has the form:
  • A.$y'+Py=Q$
  • B.$y'=g(x)h(y)$
  • C.$y''+y=0$
  • D.$(y')^2=x$
4.
After multiplying by the IF, the left side becomes:
  • A.$\dfrac{d}{dx}(y\cdot\text{IF})$
  • B.$y^2$
  • C.$\text{IF}$
  • D.$Q$
5.
The integrating factor is:
  • A.$e^{\int P\,dx}$
  • B.$\int P\,dx$
  • C.$e^{Q}$
  • D.$P\cdot Q$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Find the IF of $\dfrac{dy}{dx}+3y=x$.
7.
Find the IF of $\dfrac{dy}{dx}+\tfrac{1}{x}y=x^2$.
8.
Find the IF of $\dfrac{dy}{dx}-y=e^{x}$.
9.
Is $\dfrac{dy}{dx}+y^2=x$ a linear differential equation?
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Solve $\dfrac{dy}{dx}+\dfrac{y}{x}=x$.
11.
Solve $\dfrac{dy}{dx}+2y=e^{x}$.
12.
Solve $\dfrac{dy}{dx}+y=e^{-x}$.
13.
Solve $\dfrac{dy}{dx}+y=1$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Solve $x\dfrac{dy}{dx}+y=x^3$.
15.
Solve $\dfrac{dy}{dx}+2y\tan x=\sin x$.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $x$
  2. (B) $e^{2x}$
  3. (A) $y'+Py=Q$
  4. (A) $\dfrac{d}{dx}(y\cdot\text{IF})$
  5. (A) $e^{\int P\,dx}$
Section B — Short Answer (2 marks)
  1. $e^{3x}$.
  2. $x$.
  3. $e^{-x}$.
  4. No.
Section C — Short Answer (3 marks)
  1. $y=\tfrac{x^2}{3}+\tfrac{C}{x}$.
  2. $y=\tfrac{e^{x}}{3}+Ce^{-2x}$.
  3. $y=(x+C)e^{-x}$.
  4. $y=1+Ce^{-x}$.
Section D — Long Answer (5 marks)
  1. $y=\tfrac{x^3}{4}+\tfrac{C}{x}$.
  2. $y=\cos x+C\cos^2 x$.
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