Question

A) Find a particular solution to y" + 2y = e^3 + x^3. b) Find the general solution.

Answer 1

a.P.I=
`(e^(3x))/(11)+(1)/(2)(x^3-3x)`

b.G.S=
`C_1Cos \sqrt2 x+C_2 Sin\sqrt2 x+(1)/(11)e^(3x)+(1)/(2)(x^3-3x}`

Explanation:

We are given that a linear differential equation

`y''+2y=e^(3x)+x^3`

We have to find the particular solution

P.I=
`(e^(3x))/(D^2+2)+(x^3)/(D^2+2)`

P.I=
`(e^(3x))/(3^2+2)+(1)/(2) x^3(1+(D^2)/(2))^(-2)`

P.I=
`(e^(3x))/(11)+(1-2(D^2)/(4)+3(D^4)/(16)+...)/(2)x^3`

P.I=
`(e^(3x))/(11)+(x^3-2(\cdot3\cdot 2x)/(4))/(2)+0}` (higher order terms can be neglected

P.I=
`(e^(3x))/(11)+(1)/(2)(x^3-3x)`

b.Characteristics equation

`D^2+2=0`

`D=\pm\sqrt2 i`

C.F=
`C_1cos \sqrt2x+C_2 sin\sqrt2 x`

G.S=C.F+P.I

G.S=
`C_1Cos \sqrt2 x+C_2 Sin\sqrt2 x+(1)/(11)e^(3x)+(1)/(2)(x^3-3x)`