Find the general solution for each of the following differential equations. $x d y-\left(y+2 x^{2}\right) d x=0$

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

Find the general solution for each of the following differential equations. $x d y-\left(y+2 x^{2}\right) d x=0$

Solution:

$x d y-\left(y+2 x^{2}\right) d x=0\dots (1)$
General solution for the differential equation in the form of $\frac{d y}{d x}+P y=Q$ is given by
$y \cdot(I . F .)=\int Q \cdot(I . F \cdot) d x+c$
Where, integrating factor,
$\text { I. } F .=e \int^{p d x}$
On rearranging (1), we get
$\begin{array}{l} x d y-\left(y+2 x^{2}\right) d x=0 \\ x d y=\left(y+2 x^{2}\right) d x \\ \frac{d y}{d x}-\frac{y}{x}=2 x \ldots \ldots . .(2) \end{array}$
Equation (2) is of the form
$\frac{d y}{d x}+P y=Q$
Where, $P=-1 / x$ and $Q=2 x$
Therefore, integrating factor is
$\text { I.F. }=e \int^{p d x}=1 / \mathrm{x}$
General solution is
$\begin{array}{c} y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c \\ y \cdot \frac{1}{x}=\int 2 x \frac{1}{x} d x+C \\ \frac{y}{x}=2 x+c \end{array}$
Multiplying above equation by $\mathrm{x}$ ,
$y=2 x^{2}+c x$