Differential Equations Test

Differential Equations

This is Differential Equations Test-05 for CBSE class 12 Maths.. There are 15 questions in this test with each question having around four answer choices.

Questions & Answers

1
Which of the following is a homogeneous differential equation?
  • A
    \(\begin{array}{*{20}{l}} {\left( {4x{\text{ }} + {\text{ }}6y{\text{ }} + {\text{ }}5} \right){\text{ }}dy{\text{ }}- \\ {\text{ }}\left( {3y{\text{ }} + {\text{ }}2x{\text{ }} + {\text{ }}4} \right){\text{ }}dx{\text{ }} = {\text{ }}0} \end{array}\)
  • B
    \({y^2}dx{\text{ }} + {\text{ }}\left( {{x^2}--{\text{ }}xy{\text{ }}--{\text{ }}{y^2}} \right){\text{ }}dy{\text{ }} = {\text{ }}0\)
    Correct
  • C
    \(\left( {{x^3} + {\text{ }}2{y^2}} \right){\text{ }}dx{\text{ }} + {\text{ }}2xy{\text{ }}dy{\text{ }} = {\text{ }}0\)
  • D
    \(\begin{array}{*{20}{l}} {\left( {xy} \right){\text{ }}dx{\text{ }}--{\text{ }}\left( {{x^3} + {\text{ }}{y^3}} \right){\text{ }}dy{\text{ }} = {\text{ }}0} \end{array}\)
2
A first order linear differential equation. Is a differential equation of the form
  • A
    \(\frac{{dy}}{{dx}} + Py = Q\;\)
    Correct
  • B
    \(\frac{{dy}}{{dx}} + Px = Q\)
  • C
    \(\frac{{dy}}{{dx}} + Py = 0\;\)
  • D
    \(\frac{{dy}}{{dx}} = Q\)
3
Solution of \(\frac{{dy}}{{dx}} = \sec y\) is
  • A
    None of these
  • B
    \(x = \sin 2y + C\)
  • C
    \(x = co\operatorname{s} y + C\)
  • D
    \(x = \sin y + C\)
    Correct
4
Solution of y’ =\(\;\frac{{x + y}}{x}\) is
  • A
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}log\left| {2x} \right|{\text{ }} + Cx} \end{array}\)
  • B
    \( y=x\log \left| x \right| + Cx \)
    Correct
  • C
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}x{\text{ }}log\left| {3x} \right|{\text{ }} + Cx} \end{array}\)
  • D
    \(\begin{array}{*{20}{l}} {y{\text{ }} = {\text{ }}log\left| x \right|{\text{ }} + Cx} \end{array}\)
5
Solution of (x – y) dy – (x + y) dx = 0 is
  • A
    \(ta{n^{ - 1}}\left( {\frac{y}{x}} \right) = \;\frac{1}{2}\log \left( {{x^2} + \;{y^3}} \right) + \;C\)
  • B
    \(ta{n^{ - 1}}\left( {\frac{y}{x}} \right) = \;\frac{1}{2}\log \left( {{x^3} + \;{y^3}} \right) + \;C\)
  • C
    \(ta{n^{ - 1}}\left( {\frac{y}{x}} \right) = \;\frac{1}{2}\log \left( {{x^2} + \;{y^2}} \right) + \;C\)
    Correct
  • D
    \(ta{n^{ - 1}}\left( {\frac{y}{x}} \right) = \;\frac{1}{2}\log \left( {{x^3} + \;{y^2}} \right) + \;C\)
6
Solution of \((x + 1)\frac{{dy}}{{dx}} = 2xy\) is
  • A
    \(\log y = \left\{ {x + \log \left| x \right|} \right\} + C\)
  • B
    None of these
  • C
    \(\log y = \left\{ {x - \log \left| x \right|} \right\} + C\)
  • D
    \(\log y = 2\left\{ {x - \log \left| x \right|} \right\} + C\)
    Correct
7
Solution of differential equation \(\frac{{dy}}{{dx}} = \frac{{x + y}}{{x - y}}\) is
  • A
    \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right) = \frac{1}{2}\log ({x^3} + {y^2}) + C\)
  • B
    \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right) = \frac{1}{2}\log ({x^2} + {y^2}) + C\)
    Correct
  • C
    None of these
  • D
    \({\tan ^{ - 1}}\left( {\frac{y}{x}} \right) = \log ({x^3} + {y^2}) + C\)
8
General solution of \(\frac{{dy}}{{dx}} + \;2y = \sin x\) is
  • A
    \(\;y = \)\(\frac{1}{5}\left( {2\sin x + \cos x} \right) + C{e^{ - 2x}}\)
  • B
    \(y = \frac{1}{5}\left( {2\sin x - \cos x} \right) - C{e^{ - 2x}}\)
  • C
    \(y = \frac{1}{5}\left( {2\sin x - \cos x} \right) + C{e^{ - 2x}}\)
    Correct
  • D
    \(y = \frac{1}{5}\left( {2\sin x + \cos x} \right) - C{e^{ - 2x}}\)
9
General solution of\(\frac{{dy}}{{dx}} + \;3y = {e^{ - 2x}}\) is
  • A
    \(y = {e^{ - 2x}} + C{e^{ - 3x}}\)
    Correct
  • B
    \(y = {e^{ - 3x}} - C{e^{ - 3x}}\)
  • C
    \(y = {e^{ - 2x}} - C{e^{ - 3x}}\)
  • D
    \(y = {e^{ - 2x}} - C{e^{ - 5x}}\)
10
General solution of\(\frac{{dy}}{{dx}} + \;\frac{y}{x} = {x^2}\) is
  • A
    \(xy = \frac{{{x^7}}}{4} + C\)
  • B
    \(xy = \frac{{{x^6}}}{4} + C\)
  • C
    \(xy = \frac{{{x^5}}}{4} + C\)
  • D
    \(xy = \frac{{{x^4}}}{4} + C\)
    Correct
11
General solution of\(\frac{{dy}}{{dx}} + \left( {\sec x} \right)y = \;\tan x\;\left( {0 \leqslant x < \frac{\pi }{2}} \right)\) is
  • A
    \(y\left( {\sec x + \tan x} \right) \\ = \sec x - \tan x - x + C\)
  • B
    \(y\left( {\sec x - \tan x} \right) \\ = \sec x + \tan x - x + C\)
  • C
    \(y\left( {\sec x - \tan x} \right) \\ = \sec x - \tan x - x + C\)
  • D
    \(y\left( {\sec x + \tan x} \right) \\ = \sec x + \tan x - x + C\)
    Correct
12
General solution of\(co{s^2}x\frac{{dy}}{{dx}} + y = \;\tan x\;\left( {0 \leqslant x < \frac{\pi }{2}} \right)\) is
  • A
    \(y = \left( {\tan x - 1} \right) + C{e^{ - \tan x}}\)
    Correct
  • B
    \(y = \left( {\tan x + 1} \right) + C{e^{ - \tan x}}\)
  • C
    \(y = \left( {\tan x + 1} \right) - C{e^{ - \tan x}}\)
  • D
    \(y = \left( {\tan x - 1} \right) - C{e^{ - \tan x}}\)
13
General solution of\(x\frac{{dy}}{{dx}} + 2y = {x^2}\log x\) is
  • A
    \(y = \frac{{{x^2}}}{{16}}\left( {4\log \left| x \right| + \;1} \right) - C{x^{ - 2}}\)
  • B
    \(y = \frac{{{x^2}}}{{16}}\left( {4\log \left| x \right| + \;1} \right) + C{x^{ - 3}}\)
  • C
    \(y = \frac{{{x^2}}}{{16}}\left( {4\log \left| x \right| + \;1} \right) + C{x^{ - 2}}\)
  • D
    \(y = \frac{{{x^2}}}{{16}}\left( {4\log \left| x \right| - \;1} \right) + C{x^{ - 2}}\)
    Correct
14
General solution of\(xlogx\frac{{dy}}{{dx}} + y = \frac{2}{x}logx\) is
  • A
    \(y\log x = \frac{2}{x}\left( {1 + \log \left| x \right|} \right) + C\)
  • B
    \(y\log x = \frac{{ - 2}}{x}\left( {1 + \log \left| x \right|} \right) + C\)
    Correct
  • C
    \(y\log x = \frac{{ - 2}}{x}\left( {1 - \log \left| x \right|} \right) + C\)
  • D
    \(y\log x = \frac{2}{x}\left( {1 - \log \left| x \right|} \right) + C\)
15
General solution of\(\left( {1 + {x^2}} \right)dy + 2xy\;dx = \cot x\;dx\;(x \ne 0)\) is
  • A
    \(y = {\left( {1 + x} \right)^{ - 1}}\log \left| {\sin x} \right| - C{(1 + {x^2})^{ - 1}}\)
  • B
    \(y = {\left( {1 + x} \right)^{ - 1}}\log \left| {\sin x} \right| + C{(1 + {x^2})^{ - 1}}\)
    Correct
  • C
    \(y = {\left( {1 + x} \right)^{ - 1}}\log \left| {\sin x} \right| - C{(1 - {x^2})^{ - 1}}\)
  • D
    \(y = {\left( {1 + x} \right)^{ - 1}}\log \left| {\sin x} \right| + C{(1 - {x^2})^{ - 1}}\)