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Vidaara.orgClass 12 · Mathematics
CodeVID-M12-09-SEP-01
Variable-Separable Equations — Assignment
Chapter: Differential Equations
Topic: Variable-Separable 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.
Solving $\dfrac{dy}{dx}=\dfrac{x}{y}$ gives:
  • A.$y^2-x^2=C$
  • B.$y=Cx$
  • C.$x^2+y^2=C$
  • D.$y=x+C$
2.
The solution of $\dfrac{dy}{dx}=ky$ is:
  • A.$y=kx+C$
  • B.$y=Ce^{kx}$
  • C.$y=Cx^k$
  • D.$y=\ln(kx)$
3.
A DE is separable if it can be written as:
  • A.$\dfrac{dy}{h(y)}=g(x)dx$
  • B.$y'+Py=Q$
  • C.$y''=0$
  • D.$y=mx+c$
4.
Solving $e^{y}\,dy=e^{x}\,dx$ gives:
  • A.$e^y=e^x+C$
  • B.$y=x+C$
  • C.$e^{y}=Ce^{x}$
  • D.$y=e^x$
5.
The solution of $\dfrac{dy}{dx}=\dfrac{y}{x}$ is:
  • A.$y=Cx$
  • B.$xy=C$
  • C.$y=x+C$
  • D.$y^2=Cx$
Section B — Short Answer (2 marks) 4 × 2 = 8 marks
6.
Solve $\dfrac{dy}{dx}=2x$.
7.
Solve $\dfrac{dy}{dx}=y$.
8.
Solve $\dfrac{dy}{dx}=-\dfrac{y}{x}$.
9.
Solve $\dfrac{dy}{dx}=\sec^2 x$.
Section C — Short Answer (3 marks) 4 × 3 = 12 marks
10.
Solve $\dfrac{dy}{dx}=\dfrac{1+y^2}{1+x^2}$.
11.
Solve $\dfrac{dy}{dx}=e^{x+y}$.
12.
Solve $x\dfrac{dy}{dx}=y$.
13.
Solve $\dfrac{dy}{dx}=(1+x)(1+y)$.
Section D — Long Answer (5 marks) 2 × 5 = 10 marks
14.
Solve $(1+x^2)\dfrac{dy}{dx}=1$, given $y(0)=0$.
15.
Solve $\dfrac{dy}{dx}=y\tan x$, given $y(0)=1$.

Answer Key

Section A — Multiple Choice Questions
  1. (A) $y^2-x^2=C$
  2. (B) $y=Ce^{kx}$
  3. (A) $\dfrac{dy}{h(y)}=g(x)dx$
  4. (A) $e^y=e^x+C$
  5. (A) $y=Cx$
Section B — Short Answer (2 marks)
  1. $y=x^2+C$.
  2. $y=Ce^{x}$.
  3. $xy=C$.
  4. $y=\tan x+C$.
Section C — Short Answer (3 marks)
  1. $\tan^{-1}y=\tan^{-1}x+C$.
  2. $e^{x}+e^{-y}=C$.
  3. $y=Cx$.
  4. $\ln|1+y|=x+\tfrac{x^2}{2}+C$.
Section D — Long Answer (5 marks)
  1. $y=\tan^{-1}x$.
  2. $y=\sec x$.
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