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

1. chihiroasleaf

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

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

thank you...