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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Find the particular solution of the differential equation
$\tan x.\frac{\text{dy}}{\text{dx}}=2x \tan x+x^2-\text{y};(\tan x\neq0)\text{given that y}=0 \ \text{when x}=\frac{\pi}{2}$
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Answer

Given equation can be written as
$\Rightarrow\ \frac{\text{dy}}{\text{dx}}+(\cot\text{x})\text{y}=2\text{x}+\text{x}^2\cot\ \text{x}$
$\text{I.F.}=\text{e}^{\int\cot\text{x dx}}=\text{e}^{\log \sin\text{x}}=\sin\text{x}$
Solution is, y × sin x$=\int(2\text{x}\sin\text{x}+\text{x}^2\cos\text{x})\text{dx}$
$\Rightarrow y \sin x = x^2 \sin x + C$
$\text{When x}=\frac{\pi}{2},\text{y}=0,\text{we get c}=\frac{-\pi^2}{4}$
$\therefore\ \text{Required solution is,}\ \ \ 4\text{y}\sin\text{x}=4\text{x}^2\sin\text{x}-\pi^2$
or,$ y = x^2– \pi^2/4\  cosec\  x$

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