Question

1) Find a Real General Solution of the Following Systems

y₁'=-y₁-y₂
y₂'=y-y₂
y₁(0)=1, y₂(0)=0

2) Solve the Initial Value Problem (IVP)
y₁'=-3y₁-4y₂+5eᵗ
y₂=5y₁+6y₂-6eᵗ
y₁(0)=19, y₂(0)=-23

Answer 1

To find the real general solution of the given system of differential equations, we first need to find the eigenvalues and eigenvectors of the coefficient matrix.

Step-by-step explanation:

The given system of differential equations is:

y₁' = -y₁ - y₂

y₂' = y - y₂

We can solve this system by finding the eigenvalues and eigenvectors of the coefficient matrix. The eigenvalues are found by solving the characteristic equation:

|λ + 1 1 |

| -1 λ + 1| = 0

Using the eigenvalues, we can find the eigenvectors and use them to write the general solution of the system. Finally, we can substitute the initial values to find the particular solution.