The question: 3 solutions of a certain 2nd order non homogenous linear equation are:\psi_1

Odompombagnom6ng

Odompombagnom6ng

Answered question

2022-02-24

The question: 3 solutions of a certain 2nd order non homogenous linear equation are:
ψ1(t)=t2
ψ2(t)=t2+e2t
ψ3(t)=1+t2+2e2t
Find the general solution of the equation.
My attempt: (I closely followed an example from the book)
so ψ2(t)ψ1(t)=e2t and ψ3(t)ψ2(t)=1+e2t are solutions of the corresponding homogenous equation. Next I need to show that these are linearly independent in order to use the theorem to find the general solution. I assume they are, but I am not sure how to exactly show that?
And then using that theorem: Every solution is in the form of y(t)=c1y1(t)+c2y2(t)+ψ(t) so
y(t)=c1y1(t)+c2y2(t)+ψ(t)
y(t)=c1e2t+c2(1+e2t)+t2
y(t)=(c1+c2)e2t+c2+t2
I am not really sure about my answer, particularly during the part of assuming linear independence.

Answer & Explanation

Tatiana Berg

Tatiana Berg

Beginner2022-02-25Added 4 answers

Hint: To check for Linear independence use the Wronskian which is defined as
W[y1  y2]=det[[y1,y2],[y1,y2]]=y1y2y1y2
If the determinant is zero, what does that say about linear independence?

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