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Differential Equations

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Solving an ODE with the Laplace Transform

DIFFEQ-OKNEEN

Katie has encountered an IVP(initial value problem): $y''-3y'-10y=e^{-t}$, $y(0)=0, y'(0)=2$. She wants to solve this problem using the Laplace Transform because she thinks that this method is more straightforward than using a characteristic equation.

Assume the Laplace Transform of $y(t)$ is $Y(s)$. After some computation, Katie thinks she will obtain $Y(s)=\frac{A}{s-5}+\frac{B}{s+2}+\frac{C}{s+1}$, so that the solution is $y(t)=Ae^{5t}+Be^{-2t}+Ce^{-t}$.

However, she is having trouble finding the values of $A,B,C$. Can you help her?

A

$A=\cfrac{2}{7},B=-\cfrac{2}{7}, C=-\cfrac{1}{6}$

B

$A=\cfrac{6}{7}, B=-\cfrac{6}{7}, C=0$

C

$A=\cfrac{1}{42}, B=\cfrac{1}{7}, C=-\cfrac{1}{6}$

D

$A=\cfrac{13}{42}, B=-\cfrac{1}{7}, C=-\cfrac{1}{6}$