Question

Solve one of the following two non-homogeneous differential equations using whatever technique your prefer. Put an "X" through the equation you would not like me to grade. If you do not technique you prefer. Put an "X" through the equation you would not like me to grade. If you do not put an "X" through one of the equations, I will grade whichever problem I prefer to grade. a) y" - 4y + 4y = 6xe^2x b) y" + 9y = 5 cos x - 7 sin x

Answer 1

a.
`y(x)=c_1e^(2x)+c-2xe^(2x)+x^3e^(2x)`

b
`y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x`

Explanation:

1.
`y''-4y'+4y=6x e^(2x)`

Auxillary equation

`D^2-4D+ 4=0`

`(D-2)(D-2)=0`

D=2,2

Then complementary solution =
`C_1e^(2x)+C_2xe^(2x)`

Particular solution
`=(6 xe^(2x))/((D-2)^2)`

D is replace by D+2 then we get

P.I=
`(6xe^(2x))/(0)`

P.I=
`(e^(ax))/(D+a) \cdot .V`

where V is a function of x

P.I=
`\frac}x^3e^(2x)`

By integrating two times

Hence, the general solution

`y(x)=c_1e^(2x)+c-2xe^(2x)+x^3e^(2x)`

b.y''+9y=5 cos x-7 sin x

Auxillary equation

`D^2+9=0`

D=
`\pm 3i`

`C.F=c_1 cos 3x+ c_2sin 3x`

P.I=
`(5 cos x-7 sin x)/(D^2+9)`

P.I=
`(sin ax)/(D^2+bD +C)`

Then D square is replace by -a square

`D^2` is replace by - then we get

P.I=-5 cos x+7 sin x

The general solution

`y(x)=c_1cos 3x+c_2 sin 3x-5 cos x+ 7sin x`