Question

Determine the solution to the IVP and initial conditions
y" +4y' + 4y = (3+x)e⁻²ˣ

Answer 1

To solve the initial value problem (IVP) with the given initial conditions, use the method of undetermined coefficients to find the complementary solution and particular solution.

Step-by-step explanation:

To solve the initial value problem (IVP) with the given initial conditions, we can use the method of undetermined coefficients.

1. First, we find the complementary solution to the homogeneous equation y" + 4y' + 4y = 0.

2. Then, we find a particular solution to the nonhomogeneous equation y" + 4y' + 4y = (3 + x)e^{-2x}.

3. Finally, we combine the complementary solution and the particular solution to obtain the general solution, and use the initial conditions to find the specific solution to the IVP.

Answer 2

The solution to the given initial value problem involves solving a second-order homogeneous linear differential equation and finding a particular solution for the non-homogeneous term.

Step-by-step explanation:

The question involves finding the solution to the initial value problem (IVP) and initial conditions for a second-order linear differential equation with a non-homogeneous term. The equation given is y" +4y' + 4y = (3+x)e⁻²⁹, which is a differential equation that can be approached by first solving the homogeneous part and then finding a particular solution for the non-homogeneous part.

To solve the homogeneous equation y" +4y' + 4y = 0, one would look for solutions of the form eˣₓₓᶜ, where r needs to be determined. This leads to a characteristic equation r² + 4r + 4 = 0, which can be factored to (r+2)² = 0, yielding a double root of r = -2. Therefore, the homogeneous solution is a combination of e⁻²ₓ and xe⁻²ₓ.

For the particular solution to the non-homogeneous equation, one can use the method of undetermined coefficients or the variation of parameters. Assuming an ansatz for the particular solution of the form yp(x) = (Ax + B)e⁻²ₓ, where A and B are constants to be determined, and plugging it into the original equation would allow one to solve for A and B.