Find a particular solution satisfying the given condition for each of the following differential equations. $\frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{xy}=\mathrm{x}$, given that $y=0$ when $x=0$

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Find a particular solution satisfying the given condition for each of the following differential equations. $\frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{xy}=\mathrm{x}$ , given that $y=0$ when $x=0$

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

Find a particular solution satisfying the given condition for each of the following differential equations. $\frac{\mathrm{dy}}{\mathrm{dx}}+2 \mathrm{xy}=\mathrm{x}$ , given that $y=0$ when $x=0$

Solution:

General solution for the differential equation in the form of
$\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}\dots(1)$
General solution is given by,
$y \cdot(I . F .)=\int Q \cdot(I . F .) d x+c$
Where, integrating factor,
$\text { I. } F .=e \int^{p d x}$
Equation (1) is of the form
$\frac{d y}{d x}+\mathrm{Py}=\mathrm{Q}$
Where, $\mathrm{P}=2 \mathrm{x}$ and $\mathrm{Q}=\mathrm{x}$
Therefore, integrating factor is
$I . F .=e \int^{p d x}=e \int^{2 x d x}=e^{x^{2}}$
General solution is
$\mathrm{I}=\int 2 \mathrm{x} \cdot e^{x^{2}} \mathrm{dx}$
Put, $\mathrm{x}^{2}=\mathrm{t}=2 \mathrm{x} \mathrm{dx}=\mathrm{dt}$
$\mathrm{I}=\int e^{t} \mathrm{dt}=e^{x^{2}}$
Substituting I in (2),
$\mathrm{y} \cdot e^{x^{2}}=1 / 2 e^{x^{2}}+\mathrm{c}$ (general solution)
For particular solution put $\mathrm{y}=0$ and $\mathrm{x}=0$ in above equation, $\mathrm{c}=-1 / 2$
Substituting $\mathrm{c}$ in general solution,
$\mathrm{y} \cdot e^{x^{2}}=1 / 2 e^{x^{2}}-1 / 2$
Multiplying above equation by $\frac{2}{e^{x^{2}}}$
$2 \mathrm{y}=1-e^{-x^{2}}$ (particular solution)