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
Please solve
partial differential equations by laplace transforms,partial differential equations by laplace transforms
2yt2=a22yx29
y(x,0)=0
y(0,t)=0
yt=(x,0)=0
limxyx(x.t)=0
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Expert Answer

Jozlyn
Answered 2021-02-01 Author has 85 answers
Given problem is
2yt2=a22yx29
and given conditions are
y(x,0)=0
y(0,t)=0
yt=(x,0)=0
limxyx(x.t)=0
Step 2
Let the Laplace transformation of y is y_s then taking the Laplace transformation of equation (1)
s2yssy(x,0)yt(x,0)=a2d2ysdx29s
s2ys=a2d2ysdx29s
d2ysdx2s2a2ys=9sa2
CF of the above equation
CF=Aesax+Besax
Now PI=9s3
Hence the complete solution is
ys=Aesax+Besax+9s3
now taking the inverse Laplace transformation of above
y(x,t)=AL1(esax)+BL1(esax)+92t2
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