(a) Let the twice continuously differentiable function u(x) be a solution of the initial value problem

                                                                                                 y′′ + py^′ + qy = 0,            y(0) = 0,            y^′(0) = 1,

where p and q are constants. Show that the function

                                                                                               

is a solution of the nonhomogeneous differential equation

y^′′+ py^′+ qy = g(x),

satisfying the initial conditions y(x₀) = 0 = y^′(x₀).

Hint: Use the Leibniz’s formula.

(b) The existence and uniqueness theorem states that if p(x),q(x) and g(x) are continuous on an open interval I containing x₀, then there exists a unique solution of the initial value problem                   

 

throughout the interval I. Explain in the context of the existence and uniqueness theorem whether the function y = sin(x²) can be a solution of the initial value problem

                                                            y^′′ + p(x)y^′ + q(x)y = g(x),         y(0) = 1,            y^′(0) = −1.

(x) 3D u(x- t)g(t)dt