Question

Solve the integrodifferential equation (dy/dt) + 4y + 3∫ᵗ y dt = 16 e⁻²ᵗ u(t) , y(0) = -3.

y(t) is determine as?

Answer 1

To solve the integrodifferential equation, we can use the method of variation of parameters. The solution is y(t) = y_h(t) + y_p(t), where y_h(t) is the homogeneous solution and y_p(t) is the particular solution.

Step-by-step explanation:

To solve the integrodifferential equation, we can use the method of variation of parameters. First, let's find the homogeneous solution by solving the characteristic equation: r + 4 = 0. This gives us r = -4. So, the homogeneous solution is y_h(t) = c1e^(-4t).

For the particular solution, we assume y_p(t) = Ate^(−2t)u(t), where A is a constant to be determined. Substitute this into the equation and solve for A.

Using the initial condition y(0) = -3, we can find the value of the constant c1 in the homogeneous solution. Therefore, the solution to the integrodifferential equation is y(t) = y_h(t) + y_p(t).