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Question Bank [2022] — Maths STD 12 Science — Question

Maharashtra BoardEnglish MediumSTD 12 ScienceMathsQuestion Bank [2022]2 Marks
Question
Solve the differential equation $\frac{ d y}{ d x}= e ^{( x + y )}+ x ^2 e ^{ y }$
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Answer

$ \frac{ d y}{ d x}= e ^{( x + y )}+ x ^2 e ^{ y }$
$\therefore \frac{ d y}{ d x}= e ^{ x } \cdot e ^{ y }+ x ^2 e ^{ y }$
$\therefore \frac{ d y}{ d x}= e ^{ y }\left( e ^{ x }+ x ^2\right)$
$\therefore \frac{ d y}{ e ^y}=\left( e ^{ x }+ x ^2\right) dx $
Integrating on both sides, we get
$ \int_{ e }^{-y} d y=\int\left( e ^x+x^2\right) d x$
$\therefore \frac{ e ^{-y}}{-1}= e ^x+\frac{x^3}{3}+ c _1$
$\therefore e ^{- y }=- e ^x-\frac{x^3}{3}- c _1$
$\therefore e ^{-y}+ e ^x+\frac{x^3}{3}= c , \text { where } c = c _1 $

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