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

$$\mathrm{Consider}\mathrm{the}\mathrm{boundary}\mathrm{value}$$$$\mathrm{problem}{\mathrm{y}}^{{}^{″}}-{2\mathrm{y}}^{\prime }+2\mathrm{y}=0,\mathrm{y}\left(\mathrm{a}\right)=\mathrm{c}$$$$,\mathrm{y}\left(\mathrm{b}\right)=\mathrm{d}.$$$$1)\mathrm{If}\mathrm{this}\mathrm{problem}\mathrm{has}\mathrm{a}\mathrm{unique}$$$$\mathrm{solution},\mathrm{how}\mathrm{are}\mathrm{a}\mathrm{and}\mathrm{b}\mathrm{related}?$$$$2)\mathrm{If}\mathrm{this}\mathrm{problem}\mathrm{has}\mathrm{no}\mathrm{solution},$$$$\mathrm{how}\mathrm{are}\mathrm{a},\mathrm{b},\mathrm{c}\mathrm{and}\mathrm{d}\mathrm{related}?$$$$$$$$$$$$\mathrm{Help}!$$

Answered by FelipeLz last updated on 05/Jan/23

$$y^{″}-2y^{\prime }+2y=0$$$$y=e^r\to r^2{e}^r-2{re}^r+2e^r=0$$$$r^2-2r+2=0\to r=1\pm i$$$$y\left(x\right)=k_1{e}^x\cos \left(x\right)+k_2{e}^x\sin \left(x\right)$$$$$$$$1.y\left(a\right)=c\to k_1{e}^a\cos \left(a\right)+k_2{e}^a\sin \left(a\right)=c$$$$y\left(b\right)=d\to k_1{e}^b\cos \left(b\right)+k_2{e}^b\sin \left(b\right)=d$$$$\left|\begin{array}{cc}{e}^a\cos \left(a\right)& e^a\sin \left(a\right)\\ e^b\cos \left(b\right)& e^b\sin \left(b\right)\end{array}\right|\ne 0$$$$e^{a+b}\cos \left(a\right)\sin \left(b\right)-e^{a+b}\sin \left(a\right)\cos \left(b\right)\ne 0$$$$e^{a+b}\sin (b-a)\ne 0$$$$\sin (b-a)\ne 0$$$$b\ne a+n\pi \mathrm{\forall }n\in \mathbb{Z}$$$$$$$$2.\left|\begin{array}{cc}{e}^a\cos \left(a\right)& e^a\sin \left(a\right)\\ e^b\cos \left(b\right)& e^b\sin \left(b\right)\end{array}\right|=0$$$$e^{a+b}\cos \left(a\right)\sin \left(b\right)-e^{a+b}\sin \left(a\right)\cos \left(b\right)=0$$$$e^{a+b}\sin (b-a)=0$$$$\sin (b-a)=0$$$$b=a+n\pi \mathrm{\forall }n\in \mathbb{Z}$$$$\left|\begin{array}{cc}c& e^a\sin \left(a\right)\\ d& e^b\sin \left(b\right)\end{array}\right|\ne 0$$$${ce}^b\sin \left(b\right)-{de}^a\sin \left(a\right)\ne 0$$$${ce}^{a+n\pi }\sin (a+n\pi )-{de}^a\sin \left(a\right)\ne 0$$$${ce}^{a+n\pi }\sin \left(a\right)\cos \left(n\pi \right)-{de}^a\sin \left(a\right)\ne 0$$$$e^a\sin \left(a\right)[{ce}^{n\pi }\cos \left(n\pi \right)-d]\ne 0$$$$d\ne {ce}^{n\pi }\cos \left(n\pi \right)$$$$\{\begin{array}{l}b-a=n\pi \mathrm{\forall }n\in \mathbb{Z}\\ d\ne {ce}^{b-a}\cos (b-a)\end{array}$$$$$$

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