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

(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 and 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 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 = sin(x2) can be a solution of the initial value problem

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


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

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