Using the Annihilator Method to Solve a Differential Equation

[jbord39]

Homework Statement

Solve the given differential equation using the annihilator method:

(D-3)(D+2)y=x^2e^x

Homework Equations

D=dy/dx

The Attempt at a Solution

I think the annihilator would be (D-1)^3.

so the solution would be in the form:

y = Ae^x + Bxe^x + C(x^2)e^x + De^(3x) +Ee^(-2x)

Solving for the particular solution yields:

Ae^x+Bxe^x+C(x^2)e^x = (x^2)(e^x)

Am I solving this correctly, and if so, where do I go from here?

Thanks a bunch,

John

 

[Mark44]

Homework Statement

Solve the given differential equation using the annihilator method:

(D-3)(D+2)y=x^2e^x

Homework Equations

D=dy/dx

The Attempt at a Solution

I think the annihilator would be (D-1)^3.

so the solution would be in the form:

y = Ae^x + Bxe^x + C(x^2)e^x + De^(3x) +Ee^(-2x)

Solving for the particular solution yields:

Ae^x+Bxe^x+C(x^2)e^x = (x^2)(e^x)

Am I solving this correctly, and if so, where do I go from here?

Everything looks fine, so far.

Using your particular solution, y = Ae^x + Bxe^x + Cx^2e^x, calculate y' and y'' and substitute into your differential equation to find A, B, and C. Your differential equation is y'' - y' - 6y = x^2e^x

Your general solution will still involve undetermined coefficients for the e^(3x) and e^(-2x) terms unless you have some initial conditions.

 

[jbord39]

Thanks for the quick reply. I'll repost my final solutions.