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chihiroasleaf
Group Title
Partial Differential Equation
Solve the following boundary value problems
\[\Large \frac{\partial^{2} u}{\partial x \partial y} (x,y) =3x^{2} , u(x,0) = x^n (n > 0) , u(0,y) = 0 \]
 one year ago
 one year ago
chihiroasleaf Group Title
Partial Differential Equation Solve the following boundary value problems \[\Large \frac{\partial^{2} u}{\partial x \partial y} (x,y) =3x^{2} , u(x,0) = x^n (n > 0) , u(0,y) = 0 \]
 one year ago
 one year ago

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chihiroasleaf Group TitleBest ResponseYou've already chosen the best response.0
I've tried solve it \[ \Large \frac{\partial}{\partial x} \left(\frac{\partial u}{\partial y} (x,y) \right) = 3x^{2} \] Integrate respect to \(x\) \[ \Large \frac{\partial u}{\partial y} (x,y) = x^3 + f(y) \] Integrate respect to \(y\) \[ \Large u(x,y) = x^{3} y + F(y) + g(x) ; \frac{\partial}{\partial y} F(y) = f(y) \] \[\Large u(x,0) = x^n \implies x^{3} \cdot 0 + F(0) + g(x) = x^{n} \implies F(0) + g(x) = x^n\] \[\Large u(0,y) = 0 \implies 0 \cdot y + F(y) + g(0) = 0 \implies F(y) + g(0) = 0\] \[ \Large u(x,y) = x^3y  g(0) + x^n  F(0) \] \[ \Large u(0,y) = 0 \implies 0  g(0) + 0 F(0) = 0 \implies g(0) =  F(0) \] so.., \[ \Large u(x,y) = x^3y + x^n \] is this correct?
 one year ago

AccessDenied Group TitleBest ResponseYou've already chosen the best response.0
While I am not learning this currently, the work seems to be correct, and the answer seems to agree with your initial problem (checking by taking partial derivative).
 one year ago

chihiroasleaf Group TitleBest ResponseYou've already chosen the best response.0
thank you...
 one year ago
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