Question

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

Answer 1

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'=-8

Hence, 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=$\frac{4}{13}+\frac{3}{10}x$

$y(x)=C.F+P.I=x^2(C_1cos (3lnx)+C_2sin(3lnx))+\frac{4}{13}+\frac{3}{10}x$