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Question Bank [2022] — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Question Bank [2022]3 Marks
Question
Solve the differential equation $x \frac{ d y}{ d x}+2 y= x ^2 \log x$
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Answer

$x \frac{ d y}{ d x}+2 y= x ^2 \log x$
Dividing both sides by $x$, we get
$\frac{ d y}{ d x}+\frac{2}{x} y= x \log x$
The given equation is of the form
$\frac{ d y}{ d x}+ P y= Q \text {. }$
where $P =\frac{2}{x}$ and $Q = x \log x$
$ \therefore \text { I.F. }= e ^{\int^{ Pd x}}$
$= e ^{2 \int \frac{1}{x} d x}$
$= e ^{2 \log |x|}$
$= e ^{\log \left|x^2\right|}$
$= x ^2 $
$\therefore$ Solution of the given equation is
$ y(\text { I.F. })=\int Q (\text { I.F. }) d x+ c$
$\therefore yx ^2=\int(x \log x) x^2 d x+ c$
$=\int x^3 \log x d x+ c $
$=\log x \int x^3 d x-\int\left(\frac{ d }{ d x} \log x \int x^3 d x\right) d x+ c$
$=\log x \cdot \frac{x^4}{4}-\int \frac{1}{x}\left(\frac{x^4}{4}\right) d x+ c$
$=\frac{x^4}{4} \log x-\frac{1}{4} \int x^3 d x+ c$
$=\frac{x^4}{4} \log x-\frac{1}{4} \cdot \frac{x^4}{4}+ c$
$\therefore x ^2 y =\frac{x^4}{16}(4 \log x-1)+ c $

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