Find the general solution of x'=(2, 1, -5, -2)x+(-cos(t), sin(t)).

(this is 2x2 matrix with 2 and 1 on the left, -5 and -2 on the right. And -cos(t) on top, sin(t) on bottom. I've found the answer for the 2x2 matrix but have trouble finding it for the other part. I want to solve it using undetermined coefficients but I don't know what should the form be.

Tutors, sign in to answer this question.

You found the homogeneous solution to be a combination of terms involving sin(t) and cos(t). Since your inhomogeneous term (-cos(t), sin(t)) is of the same functional form, the inhomogeneous solution can contain terms of the form

x= (A1, A2) cos(t) + (B1, B2) sin(t) + (C1, C2) t cos(t) + (D1, D2) t sin(t)

To determine the 8 unknown coefficients A1 through D2 you would substitute this x into your original equation and get 8 linear equations.

An easier method in this case is given by variation of parameters.

Already have an account? Log in

By signing up, I agree to Wyzant’s terms of use and privacy policy.

Or

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Your Facebook email address is associated with a Wyzant tutor account. Please use a different email address to create a new student account.

Good news! It looks like you already have an account registered with the email address **you provided**.

It looks like this is your first time here. Welcome!

To present the tutors that are the best fit for you, we’ll need your ZIP code.

Please try again, our system had a problem processing your request.

## Comments

^{-1}. Then if b is the inhomogeneous term, the inhomogeneous part of the solution is given by^{-1}b dt.