Question
Using Laplace transform, solve the equation:
given that:
Answer :
Word Count : 129
Take the Laplace transform in $t$: $U(x,s)=\mathcal L\{u(x,t)\}$. Using $\mathcal L\{u_{tt}\}=s^2U-s\,u(x,0)-u_t(x,0)$ and $u_t(x,0)=0,\;u(x,0)=f(x)=10\sin(2\pi x)-20\sin(5\pi x)$, the PDE becomes $$ s^2U-sf(x)=9U_{xx}\quad\Longrightarrow\quad 9U_{xx}-s^2U=-s ________ _________ ____ ____ ____ _________ ________ ____ ____.
_____ __________ ______ ______ ___.
____ ____ ______ ____ ____ ______ ___ ____ ______ ____ ____ ___.
___ _______ ______ ___ _________ ___ ___ ____ _______.
___ ________ ___ ______ _____ ______ ______ ____.
_____ ___ ___ ___ ________ __________ ________.
____ ____ ______ __________ ________.
_________ _____ ___ ___ _____ _____ ________ __________ ___.
______ __________ _______ ___ ________ _______ ____ ______ _____.
____ _____ _______ ________ _____ _____ __________.
___ ____ ______ ______ ________ _________ _________ _____.
_______ ________ ______ ________ ____ _____ ____ ______.
______ ___ ______ ________ __________ _____ _____ _______ ______.
_________ _______ _____.
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Take the Laplace transform in $t$: $U(x,s)=\mathcal L\{u(x,t)\}$. Using $\mathcal L\{u_{tt}\}=s^2U-s\,u(x,0)-u_t(x,0)$ and $u_t(x,0)=0,\;u(x,0)=f(x)=10\sin(2\pi x)-20\sin(5\pi x)$, the PDE becomes $$ s^2U-sf(x)=9U_{xx}\quad\Longrightarrow\quad 9U_{xx}-s^2U=-s ________ _________ ____ ____ ____ _________ ________ ____ ____.
_____ __________ ______ ______ ___.
____ ____ ______ ____ ____ ______ ___ ____ ______ ____ ____ ___.
___ _______ ______ ___ _________ ___ ___ ____ _______.
___ ________ ___ ______ _____ ______ ______ ____.
_____ ___ ___ ___ ________ __________ ________.
____ ____ ______ __________ ________.
_________ _____ ___ ___ _____ _____ ________ __________ ___.
______ __________ _______ ___ ________ _______ ____ ______ _____.
____ _____ _______ ________ _____ _____ __________.
___ ____ ______ ______ ________ _________ _________ _____.
_______ ________ ______ ________ ____ _____ ____ ______.
______ ___ ______ ________ __________ _____ _____ _______ ______.
_________ _______ _____.
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