dy/dx+(2x)y = e^x I.F = e^∫2xdx = e^2logx = x^2 #160; yx^2 = ∫ e^xx^2 x^2y = x^2e^x ∫ 2xe^x = x^2e^x 2xe^x+2e^x+c #160; ...

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Particular Solution of a Differential Equation

Solve dydx+...

Question

Solve $\frac{d y}{d x} + \frac{2 y}{x} = e^{x}$ .

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Solution

$\frac{d y}{d x} + (\frac{2}{x}) y = e^{x}$
$I . F = e^{\int \frac{2}{x}} d x = e^{2 l o g x} = x^{2}$
$y x^{2} = \int e^{x} x^{2}$
$x^{2} y = x^{2} e^{x} - \int 2 x e^{x}$
$= x^{2} e^{x} - 2 x e^{x} + 2 e^{x} + c$

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