Suppose the solution of the boundary-value problem and are

Chapter 4, Problem 45E

(choose chapter or problem)

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 \(y_{1}(a)=0 \text { and } y_{2}(b)=0\). 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 \(y_{1}(b) \neq 0 \text { and } y_{2}(a) \neq 0\). Reread the assumptions following (24).]

Text Transcription:

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

y_{p}(x)=int_{a}^{b} G(x, t) f(t) d t

y_1x

y_2x

y_1a=0

y_2b=0

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

y(x)=y_px+fracBy_1b y_1x+fracAy_2a y_2x

y_1b neq 0

y_2aneq 0