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

Question

Please solve
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 (x, 0) = 0

y (0, t) = 0

\frac{\partial y}{\partial t} = (x, 0) = 0

lim_{x \to \infty} y_{x} (x . t) = 0

Answer 1

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 (x, 0) = 0

y (0, t) = 0

\frac{\partial y}{\partial t} = (x, 0) = 0

lim_{x \to \infty} y_{x} (x . t) = 0

Step 2
Let the Laplace transformation of y is y_s then taking the Laplace transformation of equation (1)

s^{2} y_{s} - s y (x, 0) - y_{t} (x, 0) = a^{2} \frac{d^{2} y_{s}}{d x^{2}} - \frac{9}{s}

\Rightarrow s^{2} y_{s} = a^{2} \frac{d^{2} y_{s}}{d x^{2}} - \frac{9}{s}

\Rightarrow \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

C F = A e^{\frac{s}{a x}} + B e^{\frac{- s}{a x}}

Now P I = \frac{9}{s^{3}}

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 (x, t) = A L^{- 1} (e^{\frac{s}{a x}}) + B L^{- 1} (e^{\frac{s}{a} x}) + \frac{9}{2} t^{2}

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