Consider the boundary value problem y^(′′) −2y′+2y=0, y(a)=c ,y(b)=d. 1) If this problem has a unique solution, how are a and b related? 2) If this problem has no solution, how are a,b,c and d related? Help!


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Question Number 184306 by Mastermind last updated on 05/Jan/23

$Consider the boundary value$ $problem y^{^{″}} - {2 y}^{'} + 2 y = 0, y (a) = c$ $, y (b) = d .$ $1) If this problem has a unique$ $solution, how are a and b related ?$ $2) If this problem has no solution,$ $how are a, b, c and d related ?$ $Help!$
	Answered by FelipeLz last updated on 05/Jan/23
	$y′′−2y′+2y = 0 y = e^r → r^2 e^r −2re^r +2e^r = 0 r^2 −2r+2 = 0 → r = 1±i y(x) = k_1 e^x cos(x)+k_2 e^x sin(x) 1. y(a) = c → k_1 e^a cos(a)+k_2 e^a sin(a) = c y(b) = d → k_1 e^b cos(b)+k_2 e^b sin(b) = d determinant (((e^a cos(a)),(e^a sin(a))),((e^b cos(b)),(e^b sin(b))))≠ 0 e^(a+b) cos(a)sin(b)−e^(a+b) sin(a)cos(b) ≠ 0 e^(a+b) sin(b−a) ≠ 0 sin(b−a) ≠ 0 b ≠ a+nπ ∀n ∈ Z 2. determinant (((e^a cos(a)),(e^a sin(a))),((e^b cos(b)),(e^b sin(b))))= 0 e^(a+b) cos(a)sin(b)−e^(a+b) sin(a)cos(b) = 0 e^(a+b) sin(b−a) = 0 sin(b−a) = 0 b = a+nπ ∀n ∈ Z determinant ((c,(e^a sin(a))),(d,(e^b sin(b))))≠ 0 ce^b sin(b)−de^a sin(a) ≠ 0 ce^(a+nπ) sin(a+nπ)−de^a sin(a) ≠ 0 ce^(a+nπ) sin(a)cos(nπ)−de^a sin(a) ≠ 0 e^a sin(a)[ce^(nπ) cos(nπ)−d] ≠ 0 d ≠ ce^(nπ) cos(nπ) { ((b−a = nπ ∀n ∈ Z)),((d ≠ ce^(b−a) cos(b−a))) :}$
	$y^{″} - 2 y^{'} + 2 y = 0$ $y = e^{r} \to r^{2} e^{r} - 2 {r e}^{r} + 2 e^{r} = 0$ $r^{2} - 2 r + 2 = 0 \to r = 1 \pm i$ $y (x) = k_{1} e^{x} \cos (x) + k_{2} e^{x} \sin (x)$ $1 . y (a) = c \to k_{1} e^{a} \cos (a) + k_{2} e^{a} \sin (a) = c$ $y (b) = d \to k_{1} e^{b} \cos (b) + k_{2} e^{b} \sin (b) = d$ $\| \begin{matrix} e^{a} \cos (a) & e^{a} \sin (a) \\ e^{b} \cos (b) & e^{b} \sin (b) \end{matrix} \| \neq 0$ $e^{a + b} \cos (a) \sin (b) - e^{a + b} \sin (a) \cos (b) \neq 0$ $e^{a + b} \sin (b - a) \neq 0$ $\sin (b - a) \neq 0$ $b \neq a + n π \forall n \in Z$ $2 . \| \begin{matrix} e^{a} \cos (a) & e^{a} \sin (a) \\ e^{b} \cos (b) & e^{b} \sin (b) \end{matrix} \| = 0$ $e^{a + b} \cos (a) \sin (b) - e^{a + b} \sin (a) \cos (b) = 0$ $e^{a + b} \sin (b - a) = 0$ $\sin (b - a) = 0$ $b = a + n π \forall n \in Z$ $\| \begin{matrix} c & e^{a} \sin (a) \\ d & e^{b} \sin (b) \end{matrix} \| \neq 0$ ${c e}^{b} \sin (b) - {d e}^{a} \sin (a) \neq 0$ ${c e}^{a + n π} \sin (a + n π) - {d e}^{a} \sin (a) \neq 0$ ${c e}^{a + n π} \sin (a) \cos (n π) - {d e}^{a} \sin (a) \neq 0$ $e^{a} \sin (a) [{c e}^{n π} \cos (n π) - d] \neq 0$ $d \neq {c e}^{n π} \cos (n π)$ ${\begin{cases} b - a = n π \forall n \in Z \\ d \neq {c e}^{b - a} \cos (b - a) \end{cases}$
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