## 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

1. chihiroasleaf Group Title

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?

2. AccessDenied Group Title

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).

3. chihiroasleaf Group Title

thank you...