Question

Solve \( x \frac{d y}{d x}+\frac{2 y}{x}=\frac{1}{x^{3}} \)

Answer 1

x\(\frac{dy}{dx}+\frac{2y}x=\frac1{x^3}\)

⇒ \(\frac{dy}{dx}+\frac{2}{x^2}y=\frac1{x^4}\) which is linear differential equation.

∴ I.F. = \(e^{\int\frac2{x^2}dx} = e^{\frac{-2}x}\)

y x I.F. = \(\int\frac1{x^4}(I.F)dx\)

⇒ y. e\(\frac{-2}x\) = \(\frac{e^{\frac2x}}{x^4}dx\)

Let \(\frac{-2}x=t\)

⇒ \(\frac2{x^2}dx=dt\)

∴ ye^{\(\frac{-2}x\)}  = \(\frac12\int\frac{t^2}4e^tdt\)

 = \(\frac18\)(t²e^t - 2te^t + e^t) + c

 = \(\frac18\)(t² - 2t + 1)e^t + c

left part

y.e^-2/x = \(\frac18\)(t - 1)²e^t + c

⇒ ye^-2/x = \(\frac18\) (-2/x - 1)²e^{-2/x + c} ( ∵ t = -2/x)

⇒ y = \(\frac18\)(2/x + 1)² + ce^2/x which is general solution of given differential equation.