User:Egm6321.f10.team3.cook/HW1

Given
Given a linear, second order ordinary differential equation with varying coefficients (L2-ODE-VC) and with boundary conditions:


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$$ y \left( a\right) =  \alpha  ;  y \left( b\right)  =   \beta $$
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From eqn. [4], pg [[media:2010_09_02_13_55_50.djvu|5-3]] of the notes, the ODE has a solution of the form:


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$$ y \left( x\right) =  c {y}^{1}_{H}\left( x\right) + d {y}^{2}_{H} \left( x\right) + {y}_{P} \left( x\right) $$     (4), pg 5-3
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Find
Find $$c,d$$ in terms of $$\alpha$$ and $$\beta$$

Solution
Boundary conditions
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$$ y \left( a\right) =  \alpha  ;  y \left( b\right)  =   \beta $$     (1)
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Now


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$$ \alpha =  c {y}^{1}_{H}\left( a\right) + d {y}^{2}_{H} \left( a\right) + {y}_{P} \left( a\right) $$     (2)
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$$ \beta =  c {y}^{1}_{H}\left( b\right) + d {y}^{2}_{H} \left( b\right) + {y}_{P} \left( b\right) $$     (3)
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Solve for $$ c $$ in eqn. (2)


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$$ \Rightarrow c = \frac{\alpha - {y}_{P} \left( a\right) - d {y}^{2}_{H} \left( a\right)}{{y}^{1}_{H}\left( a\right)} $$     (4)
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Plug (4) into (3) to eliminate $$ c $$


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$$ \beta =  \frac{{y}^{1}_{H}\left( b\right)}{{y}^{1}_{H}\left( a\right)} \left[ \alpha - {y}_{P} \left( a\right) - d {y}^{2}_{H} \left( a\right) \right] + d {y}^{2}_{H} \left( b\right) + {y}_{P} \left( b\right) $$
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For ease of notation, Let $$ \frac{{y}^{1}_{H}\left( b\right)}{{y}^{1}_{H}\left( a\right)} = \xi $$


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$$ \Rightarrow \beta = \xi\left[ \alpha - {y}_{P} \left( a\right) - d {y}^{2}_{H} \left( a\right) \right] + d {y}^{2}_{H} \left( b\right) + {y}_{P} \left( b\right) $$     (5)
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Solve (5) for $$ d $$


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$$ d = \frac{ \beta - {y}_{P} \left( b\right) + \xi \left[ {y}_{P} \left( a\right) - \alpha\right]}{{y}^{2}_{H} \left( b\right) - \xi {y}^{2}_{H} \left( a\right)} $$     (6)
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Subsitute (6) back into (4) to obtain $$ c $$


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$$ c = \frac{\alpha - {y}_{P} \left( a\right) - \left[\frac{ \beta - {y}_{P} \left( b\right) + \xi \left[ {y}_{P} \left( a\right) - \alpha\right]}{{y}^{2}_{H} \left( b\right) -  \xi {y}^{2}_{H} \left( a\right)} \right]{y}^{2}_{H} \left( a\right)}{{y}^{1}_{H}\left( a\right)} $$     (7)
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$$ c $$ and $$ d $$ can be re-writen by substituting ​$$ \frac{{y}^{1}_{H}\left( b\right)}{{y}^{1}_{H}\left( a\right)} = \xi $$

Should we sub back in here or leave as is?


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$$ c = \frac{\alpha - {y}_{P} \left( a\right) - \left[\frac{ \beta - {y}_{P} \left( b\right) + \frac{{y}^{1}_{H}\left( b\right)}{{y}^{1}_{H}\left( a\right)} \left[ {y}_{P} \left( a\right) - \alpha\right]}{{y}^{2}_{H} \left( b\right) - \frac{{y}^{1}_{H}\left( b\right)}{{y}^{1}_{H}\left( a\right)} {y}^{2}_{H} \left( a\right)} \right]{y}^{2}_{H} \left( a\right)}{{y}^{1}_{H}\left( a\right)} $$     (8)
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$$ d = \frac{ \beta - {y}_{P} \left( b\right) + \frac{{y}^{1}_{H}\left( b\right)}{{y}^{1}_{H}\left( a\right)} \left[ {y}_{P} \left( a\right) - \alpha\right]}{{y}^{2}_{H} \left( b\right) - \frac{{y}^{1}_{H}\left( b\right)}{{y}^{1}_{H}\left( a\right)}{y}^{2}_{H} \left( a\right)} $$     (9)
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