Question

y''+3y'-2y=e^-x/x

Answer 1

$y'' + 3y - 2y = \frac{ e^{-x}}{x}.\\ \text{This is a second order non-homogeneous DE}\\ \text{The characteristics equation is ;}\\ m^2+3m-2=0\\ m=\frac{-3 \pm \sqrt{3^2-4(1)(-2)}}{2(1)}\\ m=\frac{-3 \pm \sqrt{17}}{2}\\ \text{The complimentary solution is;}\\ y_c= Ae^{\frac{-3 + \sqrt{17}}{2}x} + Be^{\frac{-3 - \sqrt{17}}{2}x}.\\ \text{To get he Particular solution is;}\\ y_p=C_1y_1+C_2y_2\\ \text{Where:} ~~~~C_1=\int \frac{-y_2f(x)}{W(x)} ~~ \text{and}~~ C_2=\int \frac{y_1f(x)}{W(x)}\\ W(x)=y_1y_2'-y_1'y_2\\ W(x)=\left(\frac{-3 - \sqrt{17}}{2}x\right)e^{\frac{-3 + \sqrt{17}}{2}x}e^{\frac{-3 - \sqrt{17}}{2}x}-\left(\frac{-3 + \sqrt{17}}{2}x\right)e^{\frac{-3 + \sqrt{17}}{2}x}e^{\frac{-3 - \sqrt{17}}{2}x}=-\sqrt{17} e^{-3x}\\\\ f(x)=\frac{e^{-x}}{x}\\\\ \\C_1=\int\frac{e^{\frac{-3 + \sqrt{17}}{2}x}}{x\sqrt{17}}dx=\frac{\operatorname{Ei}{\left(\frac{ 1 - \sqrt{17}}{2}x \right)}}{\sqrt{17}}+C\\ C_2=\int\frac{e^{\frac{7 - \sqrt{17}}{2}x}}{-x\sqrt{17}}dx=-\frac{\operatorname{Ei}{\left(\frac{ 7- \sqrt{17}}{2}x \right)}}{\sqrt{17}}+C\\\\ \text{The general solution is given as;}\\ y(x)=Ay_1+By_2+C_1y_1+C_2y_2\\\\ y(x)=Ae^{\frac{-3 + \sqrt{17}}{2}x} + Be^{\frac{-3 - \sqrt{17}}{2}x}+\frac{\operatorname{Ei}{\left(\frac{ 1 - \sqrt{17}}{2}x \right)}}{\sqrt{17}}e^{\frac{-3 + \sqrt{17}}{2}x}-\frac{\operatorname{Ei}{\left(\frac{ 7- \sqrt{17}}{2}x \right)}}{\sqrt{17}}e^{\frac{-3 - \sqrt{17}}{2}x}.$