Find the general solution for each of the following differential equations. $\left(\mathrm{y}+3 \mathrm{x}^{2}\right) \frac{\mathrm{d} \mathrm{x}}{\mathrm{dy}}=\mathrm{x}$

Home » Find the general solution for each of the following differential equations.

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

Find the general solution for each of the following differential equations. $\left(\mathrm{y}+3 \mathrm{x}^{2}\right) \frac{\mathrm{d} \mathrm{x}}{\mathrm{dy}}=\mathrm{x}$

Solution:

$\left(\mathrm{y}+3 \mathrm{x}^{2}\right) \frac{d x}{d y}=\mathrm{x}-\cdots \dots(1)$
General solution for the differential equation in the form of is given by, $\frac{d y}{d x}+P y=Q$
$y \cdot\left(I . F_{.}\right)=\int Q \cdot(I . F .) d x+c$
Where, integrating factor,
$\text { I. } F .=e \int^{p d x}$
On rearranging $(1)$
$\begin{array}{l} \left(y+3 x^{2}\right) \frac{d x}{d y}=x \\ \frac{d y}{d x}-\frac{y}{x}=3 x \ldots \ldots \ldots(2) \end{array}$
Equation (2) is of the form
$\frac{d y}{d x}+P y=Q$
Where, $\mathrm{P}=-1 / \mathrm{x}$ and $\mathrm{Q}=3 \mathrm{x}$
Therefore, integrating factor is
$\text { I. F. }=e \int^{p d x}=\frac{1}{x}$
General solution is
$y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$
$y \cdot \frac{1}{x}=\int 3 x \cdot(1 / x) d x+c$
$\frac{y}{x}=3 x+c$
Multiplying above equation by $\mathrm{x}$ ,
$y=3 x^{2}+c x$