Find the general solution for each of the following differential equations. $x \frac{d y}{d x}+2 y=x^{2} \log x$

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CBSE | Class 12 | Linear Differential Equations | Maths | RS Aggarwal

Find the general solution for each of the following differential equations. $x \frac{d y}{d x}+2 y=x^{2} \log x$

Solution:

$\begin{array}{r} x \frac{d y}{d x}+2 y=x^{2} \log x-\ldots(1) \\ \qquad \frac{d(\log x)}{d x}=\frac{1}{x} \end{array}$
General solution for the differential equation in the form of
$\frac{d y}{d x}+P y=Q$
General solution is given by,
$\begin{array}{c} \text { y. }(I . F .)=\int Q \cdot(I . F .) d x+c \text { Where, integrating factor, } \\ \qquad I . F .=e \int p d x \end{array}$
Dividing equation (1) by x,
$\frac{d y}{d x}+\frac{2}{x} y=x \log x \ldots \ldots . .(2)$
Comparing (2) with
$\frac{d y}{d x}+P y=Q$
Where, $\mathrm{P}=2 / \mathrm{x}$ and $\mathrm{Q}=\mathrm{x}$ log $\mathrm{x}$
Therefore, integrating factor is
$\text { I.F. }=e \int^{p d x}=x^{2}$
General solution is
$\begin{array}{r} y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c \\ y\left(x^{2}\right)=\int x^{2} \cdot x \log x d x+c \ldots \ldots(3) \end{array}$
Let, $\mathrm{u}=\log \mathrm{x}$ and $\mathrm{v}=\mathrm{x}^{3}$
$\begin{array}{l} \therefore \mathrm{I}=\log \mathrm{x} \int x^{3} d x-\int\left(\frac{d}{d x}(\log x) \cdot \int x^{3} d x\right) d x \\ \therefore I=\frac{x^{4}}{4} \cdot \log x-\frac{x^{4}}{16} \end{array}$
Substituting I in (3),
$\mathrm{xy}=\frac{x^{4}}{4} \log x-\frac{x^{4}}{16}+c$
Dividing above equation by $\mathrm{x}^{2}$ ,
$y=\frac{x^{2}}{16}(1 \cdot \log x-1)+\frac{c}{x^{2}}$