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9. differential equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMaths9. differential equations1 Mark
MCQ
The solution of the differential equation $\frac{{dy}}{{dx}} + 2y\cot x = 3{x^2}{\rm{cose}}{{\rm{c}}^2}x$ is
  • ✓
    $y {\sin ^2}x = {x^3} + c$
  • B
    $y\sin x = c$
  • C
    $y\cos {x^2} = c$
  • D
    $y\sin {x^2} = c$
✓

Answer

Correct option: A.
$y {\sin ^2}x = {x^3} + c$
a
(a) $\frac{{dy}}{{dx}} + 2\cot x.y = 3{x^2}{\rm{cose}}{{\rm{c}}^2}x$

This is a linear differential equation in $y.$

$I.F.$$ = {e^{2\int_{}^{} {\cot xdx} }} = {e^{2\log \sin x}} = {\sin ^2}x$

$y. (I.F.)=$$\int_{}^{} {Q({\rm{I}}{\rm{.F}}{\rm{.}}){\rm{ }}dx} $

$y.{\sin ^2}x = \int_{}^{} {3{x^2}{\rm{cose}}{{\rm{c}}^2}x.{{\sin }^2}xdx = {x^3} + c} $.

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