# PDE:Integration and Separation of Variables

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solution $3u_{x}+4u_{y}-2u=1,u(x,0)=e^{x}\,$ solution $z_{xy}=x^{2}y,z(x,0)=x^{2},z(1,y)=\cos(y)\,$ solution $z_{xy}={\frac {1}{2}}xy^{2},z(x,0)=e^{x},z(0,y)=\sin(y)\,$ solution $u_{xx}-u_{t}=0\,$ solution $u_{t}=ku_{xx}\,$ $u(x,0)=f(x)\,$ $u(0,t)=0\,$ $u(l,t)=0\,$ $u(x,t)={\frac {2}{l}}\sum _{n=1}^{\infty }A_{n}e^{-\left({\frac {n\pi }{l}}\right)^{2}kt}\sin({\frac {n\pi x}{l}})\,$ solution $u_{t}=ku_{xx}\,$ $u_{x}(0,t)=0\,$ $u_{x}(1,t)=0\,$ $u(x,0)=\phi (x)\,$ $u(x,t)=\sum _{n=1}^{\infty }A_{n}e^{-\lambda _{n}t}\cos({\sqrt {\lambda _{n}}}x)\,$ solution Transform this initial boundary value problem into one with homogeneous boundary conditions.

 $u_{t}=u_{xx}+w(x,t)\,$ $u_{x}(0,t)=\alpha (t)\,$ $u_{x}(1,t)=\beta (t)\,$ $u(x,0)=\phi (x)\,$ $0 $t>0\,$ solution $u_{tt}=c^{2}(u_{xx}+u_{yy})\,$ $u(x,0,t)=0\,$ $u(x,b,t)=0\,$ $u(0,y,t)=0\,$ $u(a,y,t)=0\,$ $u(x,y,0)=f(x,y)\,$ $u_{t}(x,y,0)=g(x,y)\,$ $00\,$ • $u(x,y,t)=\sum _{m,n=1}^{\infty }\sin({\frac {m\pi x}{a}})\sin({\frac {n\pi y}{b}})\left[A_{m,n}\cos({\sqrt {\lambda _{m,n}}}\,ct)+B_{m,n}\sin({\sqrt {\lambda _{m,n}}}\,ct)\right]\,$ solution $u_{t}=k(\Delta u)+q(x,y,t)\,$ $u(x,0,t)=0\,$ $u(x,b,t)=0\,$ $u(0,y,t)=0\,$ $u(a,y,t)=0\,$ $u(x,y,0)=f(x,y)\,$ $00\,$ solution Transform this equation: $u_{t}=\nu u_{xx}+\lambda u_{x}+\mu u\,$ into the standard heat equation: $v_{t}=v_{xx}\,$ solution $u_{x}+2u_{y}=0,\,\,\,u(0,y)=3e^{-2y}\,$ solution $u_{xx}=a^{-2}u_{t}\,$ $u(0,t)=10\,$ $u(10,t)=30\,$ $u(x,0)=0\,$ $00\,$ $u(x,t)=2x+10+{20 \over \pi }\sum _{n=1}^{\infty }{3(-1)^{n}-1 \over n}\sin({n\pi x \over 10})e^{-a^{2}{n^{2}\pi ^{2} \over 10^{2}}t}\,$ Solve Dirichlet's problem for a circular annulus. The domain $D\,$ is the space between two concentric circles, $C_{1}$ being the innermost circle with radius $a$ , and $C_{2}$ being the outermost circle with radius $b$ . $\nabla ^{2}u=0\,$ in $D\,$ $u=g\,$ on $C_{1}\,$ $u=f\,$ on $C_{2}\,$ • $u(r,\theta )={\frac {1}{2}}(A_{0}+B_{0}\log r)+\sum _{n=1}^{\infty }\left[(A_{n}r^{n}+B_{n}r^{-n})\cos n\theta +(C_{n}r^{n}+D_{n}r^{-n})\sin n\theta \right]\,$ 