Use an appropriate infinite series method about x=0 to find two solutions of the given differential equation. y” – x2y’ + xy =0 Solve the given initial-value problem. (x+2)y” +3y=0, y(0)=0, y’(0)=1...

Answer each problem: Use an appropriate infinite series method about x=0 to find two solutions of the given differential equation. y” – x2y’ + xy =0 Solve the given initial-value problem.(x+2)y” +3y=0, y(0)=0, y’(0)=1 Use the definition of the Laplace transform to find ?{f(t)} f(t) = t, 0=t=1 2-t, t=1 Use the Laplace transform to solve the given equation. Y”-8y’ +20y = tet, y(0)=0, y’(0)=0 Solve the given linear system. dxdt = -4x + 2y dydt = 2x – 4y Solve the given linear system. x’ = -2 5-2 4 x Consider the linear system X’ = AX of two differential equations, where A is a real coefficient matrix. What is the general solution of the system if it is known that ? = 1 +2i is an eigenvalue and K1 = 1i is a corresponding eigenvector? Use Euler’s method to approximate y(0.2), where y(x) is the solution of the initial-value problem y” – (2x +1)y = 1, y(0)=3, y’(0) =1. First use one step with h=0.2 and then calculations using two steps with h=0.1 Use the finite difference method with n=10 to approximate the solution of the boundary-value problem y” + 6.55(1+x)y = 1, y(0)=0, y(1) = 0. Without actually solving the differential equation (1-2sinx) y” + xy = 0, find a lower bound for the radius of convergence of power series solutions about the ordinary point x=0.