User:Egm4313.s12.team13.akm.r2

Problem R2.5a
Problem 16 on pg 59-Find an ODE $$ \displaystyle y'' + ay' +by=0$$ for the given basis of  $$ \displaystyle e^{2.6x}, e^{-4.3x}$$ This is a case with two distinct Real-Roots. Therefore the general equation for the ODE will be


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$$  \displaystyle y=c_1e^{\lambda_1x}+c_2e^{\lambda_2x} $$     (5.0a)
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Where $$\displaystyle {\lambda_1x}=2.6x $$  and   $$ \displaystyle {\lambda_2x} = -4.3x $$ so that


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$$  \displaystyle y=c_1e^{2.6}+c_2e^{-4.3} $$     (5.1a)
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The auxiliary equation is therefore


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$$  \displaystyle ({\lambda}-2.6)({\lambda}+4.6)=0 $$     (5.2a)
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$$  \displaystyle {\lambda^2} + 1.7{\lambda}-11.18=0 $$     (5.3a)
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which results in


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$$  \displaystyle y'' + 1.7y'-11.18y=0 $$     (5.4a)
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Solution

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$$\displaystyle y'' + 1.7y'-11.18y=0 $$
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Section 1 Lecture Notes