Find the Variables

Problem

x*e(2*t)*[[1],[0]]+y*e(2*t)*[[0],[−1]]+t*[[1],[0]]=[[0],[1]]

Solution

1. Distribute the scalars into the vectors to express the equation as a single vector sum.

[[x*e(2*t)],[0]]+[[0],[−y*e(2*t)]]+[[t],[0]]=[[0],[1]]

2. Combine the vectors on the left side into a single vector by adding the corresponding components.

[[x*e(2*t)+t],[−y*e(2*t)]]=[[0],[1]]

3. Create a system of linear equations by equating the corresponding components of the vectors.

x*e(2*t)+t=0

−y*e(2*t)=1

4. Solve the first equation for x by subtracting t and dividing by e(2*t)

x*e(2*t)=−t

x=−t*e(−2*t)

5. Solve the second equation for y by dividing by −e(2*t)

y=−1/(e(2*t))

y=−e(−2*t)

Final Answer

x=−t*e(−2*t),y=−e(−2*t)

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