Final answer:
To solve the given initial value problem, we can use the methods of undetermined coefficients and variation of parameters. The particular solution can be found by guessing a specific form and solving for the constant. The general solution of the homogeneous equation and the general solution of the nonhomogeneous equation can then be determined.
Step-by-step explanation:
To solve the initial value problem y′′-4y′+4y=e⁻²ˣ/1+x² with the initial condition y(0)=5, we can use the method of undetermined coefficients and the method of variation of parameters. The particular solution can be found by guessing a particular solution in the form of Ae⁻²ˣ/(1+x²), where A is a constant to be determined. The general solution of the homogeneous equation y′′-4y′+4y=0 is y(x) = C₁e^(2x) + C₂xe^(2x), where C₁ and C₂ are arbitrary constants.
To determine the value of A, we substitute y=Ae⁻²ˣ/(1+x²) into the differential equation and solve for A. Once we find A, the general solution of the nonhomogeneous equation can be written as y(x) = C₁e^(2x) + C₂xe^(2x) + Ae⁻²ˣ/(1+x²). Using the initial condition y(0)=5, we substitute x=0 and y=5 into the general solution to find A.