Assistance with Third Order Differential Equation

[kelvin2013]

Homework Statement

Hi,

Just wondering if anyone knows how to solve the following as I am not sure where to start at all:

y''' + 8y = xsin(2x)

Any help would be great.

Homework Equations

The Attempt at a Solution

I'm thinking solving the homogeneous DE y'''+8y = 0 and then as far as I can see a possible solution maybe of the form yp=Axsin(2x)+Bxcos(2x)+Csin(2x)+Dcos(2x) ?

 

[LCKurtz]

Homework Statement

Hi,

Just wondering if anyone knows how to solve the following as I am not sure where to start at all:

y''' + 8y = xsin(2x)

Any help would be great.

Homework Equations

The Attempt at a Solution

I'm thinking solving the homogeneous DE y'''+8y = 0 and then as far as I can see a possible solution maybe of the form yp=Axsin(2x)+Bxcos(2x)+Csin(2x)+Dcos(2x) ?

What do you get for $y_c$ when you solve the homogeneous DE? You need to know that before you can predict the form of the NH equation.

 

[kelvin2013]

Hi,

thanks for the reply - I get:

y_p=c₁e^-2x+c₂e^(1+i√3)x+c₃e^(1-i√3)x

?

 

[LCKurtz]

Hi,

thanks for the reply - I get:

y_p=c₁e^-2x+c₂e^(1+i√3)x+c₃e^(1-i√3)x

?

OK. So far so good. It would be nicer to use sines and cosines instead of the complex exponentials. Anyway, your complementary solution doesn't contain any terms like the NH term. So what happens when you try your proposed $y_p$?