Final answer:
The differential equation (d²y/dx²) + (x² + 1)y = x²cos(x) is solved by finding the complimentary solution using a series solution and the particular solution using the method of undetermined coefficients. The general solution is obtained by combining these solutions and applying the initial conditions.
Step-by-step explanation:
The differential equation given is (d²y/dx²) + (x²+1)y = x²cos(x), with initial conditions y(0)=1 and y'(0)=1. To solve this differential equation, we look for a complimentary solution using a series solution around x=0. For the particular solution, we apply the method of undetermined coefficients.
First, let's find the complimentary solution y₁(x). This part of the solution will address the homogeneous part of the differential equation, (d²y/dx²) + (x²+1)y = 0. We solve it around x=0 using a power series expansion.
Next, we need to find a particular solution y₂(x) to the non-homogeneous equation. Since the non-homogeneous part is x²cos(x), a good guess for the particular solution would be a polynomial times a cosine function. We assume a solution of the form Ax²cos(x) + Bxsin(x), where A and B are coefficients to be determined.
After finding both y₁(x) and y₂(x), we combine them to form the general solution y(x) = y₁(x) + y₂(x). We then use the initial conditions to solve for the constants in the complimentary solution and the coefficients A and B in the particular solution.