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

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

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

Solution:

$x \frac{d y}{d x}-y=2 x^{2} \sec x \dots (1)$
General solution for the differential equation in the form of
$\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$
General solution is given by,
$y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$
Where, integrating factor,
$\begin{array}{r} \text { I. } F .=e \int p d x \\ \mathrm{x} \frac{d y}{d x}-\mathrm{y}=2 x^{2} \sec x \ldots \ldots \ldots(1) \end{array}$
Dividing above equation by $x$ ,
$\therefore \frac{d y}{d x}-\frac{1}{x} \mathrm{y}=2 \mathrm{xsec} x$
Equation (1) is of the form
$\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$
Where, $\mathrm{P}=\frac{-1}{x}$ and $\mathrm{Q}=2 \mathrm{x} \sec x$
Therefore, integrating factor is
$\text { I. } F .=e \int^{p d x}=\frac{1}{x}$
General solution is
$\begin{array}{l} y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c \\ \therefore \mathrm{y} \cdot\left(\frac{1}{x}\right)=\int(2 x \sec x) \cdot\left(\frac{1}{x}\right) \mathrm{d} x+\mathrm{c} \\ \therefore \mathrm{y} \cdot\left(\frac{1}{x}\right)=2 \log \mid \sec x+\tan x_{\mid}+\mathrm{c} \end{array}$
Multiplying above equation by $x$ ,
$\therefore y=2 x \log |\sec x+\tan x|+c x$