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

a.
`y(x)=C_1x^(-1)+C_2x^(-8)+(1)/(30)x^2`

b.
`y(x)=x^2(C_1cos (3lnx)+C_2sin(3lnx))+(4)/(13)+(3)/(10)x`

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(1)/(x)+C_2(1)/(x^8)`

P.I=
`(e^(2t))/(D'^2+9D'+8)=(e^(2t))/(4+18+8)`

Where D'=2

`P.I=(1)/(30)e^(2t)=(1)/(30)x^2`

`y(x)=C_1x^(-1)+C_2x^(-8)+(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=
`(4e^(0t))/(D'^2-4D'+13)+3(e^t)/(D'^2-4D'+13)`

Substitute D'=0 where
`e^(0t)` and D'=1 where
`e^t`

P.I=
`(4)/(13)+(3)/(10)e^t`

P.I=
`(4)/(13)+(3)/(10)x`

`y(x)=C.F+P.I=x^2(C_1cos (3lnx)+C_2sin(3lnx))+(4)/(13)+(3)/(10)x`