 # Please solve partial differential equations by laplace transforms,partial differential equations by laplace transforms frac{partial^2y}{partial t^2}=a^2frac{partial^2y}{partial x^2}-9 y(x,0)=0 y(0,t)=0 frac{partial y}{partial t}=(x,0)=0 lim_{xrightarrowinfty}y_x(x.t)=0 Anish Buchanan 2021-01-31 Answered
partial differential equations by laplace transforms,partial differential equations by laplace transforms
$\frac{{\partial }^{2}y}{\partial {t}^{2}}={a}^{2}\frac{{\partial }^{2}y}{\partial {x}^{2}}-9$
$y\left(x,0\right)=0$
$y\left(0,t\right)=0$
$\frac{\partial y}{\partial t}=\left(x,0\right)=0$
$\underset{x\to \mathrm{\infty }}{lim}{y}_{x}\left(x.t\right)=0$
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Given problem is
$\frac{{\partial }^{2}y}{\partial {t}^{2}}={a}^{2}\frac{{\partial }^{2}y}{\partial {x}^{2}}-9$
and given conditions are
$y\left(x,0\right)=0$
$y\left(0,t\right)=0$
$\frac{\partial y}{\partial t}=\left(x,0\right)=0$
$\underset{x\to \mathrm{\infty }}{lim}{y}_{x}\left(x.t\right)=0$
Step 2
Let the Laplace transformation of y is y_s then taking the Laplace transformation of equation (1)
${s}^{2}{y}_{s}-sy\left(x,0\right)-{y}_{t}\left(x,0\right)={a}^{2}\frac{{d}^{2}{y}_{s}}{d{x}^{2}}-\frac{9}{s}$
$⇒{s}^{2}{y}_{s}={a}^{2}\frac{{d}^{2}{y}_{s}}{d{x}^{2}}-\frac{9}{s}$
$⇒\frac{{d}^{2}{y}_{s}}{d{x}^{2}}-\frac{{s}^{2}}{{a}^{2}}{y}_{s}=\frac{9}{s{a}^{2}}$
CF of the above equation
$CF=A{e}^{\frac{s}{ax}}+B{e}^{\frac{-s}{ax}}$

Hence the complete solution is
${y}_{s}=A{e}^{\frac{s}{a}x}+B{e}^{\frac{-s}{a}x}+\frac{9}{{s}^{3}}$
now taking the inverse Laplace transformation of above
$y\left(x,t\right)=A{L}^{-1}\left({e}^{\frac{s}{ax}}\right)+B{L}^{-1}\left({e}^{\frac{s}{a}x}\right)+\frac{9}{2}{t}^{2}$