Final answer:
To find the solution to the initial value problem, we first solve the homogeneous equation and find the homogeneous solution. Then, we use the method of variation of parameters to find a particular solution to the non-homogeneous equation. By applying the initial conditions, we determine the values of the constants and obtain the solution.
Step-by-step explanation:
To find the solution to the initial value problem, we first need to solve the homogeneous equation y''+4y'+4y=0. The characteristic equation is r^2+4r+4=0, which factors as (r+2)(r+2)=0. So, the homogeneous solution is y_h=C_1e^(-2t)+C_2te^(-2t).
Next, we need to find a particular solution to the non-homogeneous equation y''+4y'+4y=(3+x)e^(-2x). We can use the method of variation of parameters to find the particular solution. Let's assume the particular solution has the form y_p=u_1(t)e^(-2t)+u_2(t)te^(-2t). By substituting this into the equation, we can solve for u_1'(t) and u_2'(t) to find their values.
Then, by applying the initial conditions y(0)=2 and y'(0)=5, we can determine the values of C_1 and C_2 in the homogeneous solution, and the values of u_1(t) and u_2(t) in the particular solution. Finally, the solution to the initial value problem is y(t)=y_h(t)+y_p(t).