Using exponential function to solve a DE

[James889]

Hi,

I have the equation $$y'' +4y = t~sin(t)$$

i know that you usually guess the solution by substituting y for a polynomial (or whatever the form of the right side is).

But i want to do this by using the exponential function exp.

so, set y to equal $$te^{it}$$
chain rule:$$y^{\prime} = 1 \cdot e^{it} +t \cdot ie^{it} = e^{it} +tie^{it}$$

$$y^{\prime\prime} = ie^{it} + ie^{it} \cdot ti + ie^{it} = 2ie^{it} \cdot ti + ie^{it}$$

hm, is this really correct?
and how would you proceed?

 

[kurt.physics]

Yes, your calculations are correct. To proceed, you would need to substitute the expression for y and its derivatives into the differential equation and solve for t. This will give you a solution of the form y=te^{it}.