Suppose the solution of the boundary-value problem y0 Py9 Qy f(x), y(a) 0, y(b) 0, a

Chapter 3, Problem 45

(choose chapter or problem)

Discussion Problems

Suppose the solution of the boundary-value problem

\(y^{\prime \prime}+P y^{\prime}+Q y=f(x)\),  y(a)=0,  y(b)=0,

a<b, is given by \(y_{p}(x)=\int_{a}^{b} G(x, t) f(t) d t\) where \(y_{1}(x)\) and \(y_{2}(x)\) are solutions of the associated homogeneous differential equation chosen in the construction of G(x, t) so that \(y_{1}(a)=0\) and \(y_{2}(b)=0\). Prove that the solution of the boundary-value problem with nonhomogeneous DE and boundary conditions,

\(y^{\prime \prime}+P y^{\prime}+Q y=f(x)\), y(a)=A, y(b)=B

is given by

\(y(x)=y_{p}(x)+\frac{B}{y_{1}(b)} y_{1}(x)+\frac{A}{y_{2}(a)} y_{2}(x)\)

[Hint: In your proof, you will have to show that \(y_{1}(b) \neq 0\) and \(y_{2}(a) \neq 0\). Reread the assumptions following (23).]

Unfortunately, we don't have that question answered yet. But you can get it answered in just 5 hours by Logging in or Becoming a subscriber.

Becoming a subscriber
Or look for another answer

×

Login

Login or Sign up for access to all of our study tools and educational content!

Forgot password?
Register Now

×

Register

Sign up for access to all content on our site!

Or login if you already have an account

×

Reset password

If you have an active account we’ll send you an e-mail for password recovery

Or login if you have your password back