Q:

# Solve one of the following non-homogeneous Cauchy-Euler equations using whatever technique you prefer. Put an "X" through whichever equations you would not put an "X" through either equation, I will grade whichever one I prefer. a) x^2 y" + 10xy' + 8y = x^2 b) x^2 y" - 3xy' + 13y = 4 + 3x

Accepted Solution

A:
Answer:a.$$y(x)=C_1x^{-1}+C_2x^{-8}+\frac{1}{30}x^2$$b.$$y(x)=x^2(C_1cos (3lnx)+C_2sin(3lnx))+\frac{4}{13}+\frac{3}{10}x$$Step-by-step explanation:1.$$x^2y''+10xy'+8y =x^2$$It is Cauchy-Euler equation where $$x=e^t$$Then auxillary equation$$D'(D'-1)+10D'+8=0$$$$D'^2+9D'+8=0$$$$(D'+1)(D'+8)=0$$D'=-1 and D'=-8Hence, C.F=$$C_1e^{-t}+C_2e^{-8t}$$C.F=$$C_1\frac{1}{x}+C_2\frac{1}{x^8}$$P.I=$$\frac{e^{2t}}{D'^2+9D'+8}=\frac{e^{2t}}{4+18+8}$$ Where D'=2$$P.I=\frac{1}{30}e^{2t}=\frac{1}{30}x^2$$$$y(x)=C_1x^{-1}+C_2x^{-8}+\frac{1}{30}x^2$$b.$$x^2y''-3xy'+13y=4+3x$$Same method apply Auxillary equation$$D'^2-D'-3D'+13=0$$$$D'^2-4D'+13=0$$$$D'=2\pm3i$$C.F=$$e^{2t}(C_1cos 3t+C_2sin 3t)$$C.F=$$x^2(C_1cos (3lnx)+C_2sin(3lnx))$$$$e^t=x$$P.I=$$\frac{4e^{0t}}{D'^2-4D'+13}+3\frac{e^t}{D'^2-4D'+13}$$Substitute D'=0 where$$e^{0t}$$ and D'=1 where $$e^t$$P.I=$$\frac{4}{13}+\frac{3}{10}e^t$$P.I=$$\frac{4}{13}+\frac{3}{10}x$$$$y(x)=C.F+P.I=x^2(C_1cos (3lnx)+C_2sin(3lnx))+\frac{4}{13}+\frac{3}{10}x$$