Question

Solve the Initial Value Problem:

y_{1}^{\prime} = -3 y_{1} - 4 y_{2} + 5 e^{t} \\
y_{2}^{\prime} = 5 y_{1} + 6 y_{2} - 6 e^{t} \\
y_{1}(0) = 19, \quad y_{2}(0) = -23
\end{array} \]
Please provide the step-by-step solution for this initial value problem.

Answer 1

To solve the given Initial Value Problem, we can use the method of solving systems of linear differential equations. We can rewrite the equations in matrix form, find the eigenvalues and eigenvectors of the matrix, and express the solution using them.

Step-by-step explanation:

To solve the given Initial Value Problem, we can use the method of solving systems of linear differential equations. We have two equations:

`y1' = -3y1 - 4y2 + 5e^t`

`y2' = 5y1 + 6y2 - 6e^t`

We can rewrite these equations in matrix form as:

Y' = AY + F

where Y = [y1, y2] and A = [[-3, -4], [5, 6]].

To solve this system, we can use the method of finding the eigenvalues and eigenvectors of the matrix A.

By finding the eigenvalues and eigenvectors of A, we can express the solution as:

`Y(t) = c1 * v1 * e^(lambda1 t) + c2 * v2 * e^(lambda2 t) + Yp(t)`

where c1 and c2 are constants, v1 and v2 are the eigenvectors of A, lambda1 and lambda2 are the eigenvalues of A, and Yp(t) is a particular solution.