View source for PDE:Integration and Separation of Variables

[[PDE1|solution]] <math>3u_x+4u_y-2u=1, u(x,0)=e^x\,</math><br><br>

[[PDE2|solution]] <math>z_{xy} = x^2y, z(x,0)=x^2, z(1,y)=\cos(y)\,</math><br><br>

[[PDE3|solution]] <math>z_{xy} = \frac{1}{2}xy^2, z(x,0)=e^x, z(0,y)=\sin(y)\,</math><br><br>
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[[PDE4|solution]] <math>u_{xx}-u_t=0\,</math><br><br>


<table border=0 cellspacing=0 width=70%><tr><td valign=top>[[PDE5|solution]] <math>u_t=ku_{xx}\,</math></td><td valign=top width=50%><math>u(x,0) = f(x)\,</math><br><math>u(0,t) = 0\,</math><br><math>u(l,t) = 0\,</math><br>
*<math>u(x,t) = 
\frac{2}{l} \sum_{n=1}^\infty A_n e^{-\left(\frac{n\pi}{l}\right)^2 k t} \sin(\frac{n\pi x}{l})\,</math><br><br>

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<table border=0 cellspacing=0 width=70%><tr><td valign=top width=50%>[[PDE6|solution]] <math>u_t=ku_{xx}\,</math></td><td valign=top width=50%><math>u_x(0,t) = 0\,</math><br><math>u_x(1,t) = 0\,</math><br><math>u(x,0) = \phi(x)\,</math><br>
*<math>u(x,t) = \sum_{n=1}^\infty A_n e^{-\lambda_n t} \cos(\sqrt{\lambda_n}x)\,</math><br><br>
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[[PDE6.5|solution]] Transform this initial boundary value problem into one with homogeneous boundary conditions.<br> 
<table border=0 cellspacing=0 width=70%><tr><td valign=top width=50%> <math>u_t=u_{xx} + w(x,t)\,</math></td><td valign=top width=50%><math>u_x(0,t) = \alpha(t)\,</math><br><math>u_x(1,t) = \beta(t)\,</math><br><math>u(x,0) = \phi(x)\,</math><br>
<math>0<x<1\,</math><br>
<math>t>0\,</math><br>
</td></tr></table><br><br>



<table border=0 cellspacing=0 width=70%><tr><td valign=top width=50%>[[PDE7|solution]] <math>u_{tt}=c^2(u_{xx}+u_{yy})\,</math></td><td valign=top width=50%>

<math>u(x,0,t) = 0\,</math><br>
<math>u(x,b,t) = 0\,</math><br>
<math>u(0,y,t) = 0\,</math><br>
<math>u(a,y,t) = 0\,</math><br>
<br>
<math>u(x,y,0) = f(x,y)\,</math><br>
<math>u_t(x,y,0) = g(x,y)\,</math><br>
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<math>0<x<a,\,\,\,\,\, 0<y<b,\,\,\,\,\, t>0\,</math><br>

</td></tr></table><br><br>
*<math>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}}\,c t) + B_{m,n} \sin(\sqrt{\lambda_{m,n}}\,c t)\right]\,</math><br><br>


<table border=0 cellspacing=0 width=70%><tr><td valign=top width=50%>[[PDE8|solution]] <math>u_{t}=k(\Delta u) + q(x,y,t)\,</math></td><td valign=top width=50%>

<math>u(x,0,t) = 0\,</math><br>
<math>u(x,b,t) = 0\,</math><br>
<math>u(0,y,t) = 0\,</math><br>
<math>u(a,y,t) = 0\,</math><br>
<br>
<math>u(x,y,0) = f(x,y)\,</math><br>
<br>
<math>0<x<a,\,\,\,\,\, 0<y<b,\,\,\,\,\, t>0\,</math><br>
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[[PDE9|solution]] Transform this equation: <math>u_{t}=\nu u_{xx} + \lambda u_x + \mu u\,</math> into the standard heat equation: <math>v_t=v_{xx}\,</math><br><br>

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[[PDE10|solution]] <math>u_x + 2u_y = 0,\,\,\,u(0,y) = 3e^{-2y}\,</math><br><br>


[[PDE11|solution]] <math>u_{xx}=a^{-2}u_t\,</math><br><br> 
<table border=0 cellspacing=0 width=100%><tr><td valign=top width=30%>
</td><td valign=top width=70%>

<math>u(0,t) = 10\,</math><br>
<math>u(10,t) = 30\,</math><br>
<br>
<math>u(x,0) = 0\,</math><br>
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<math>0<x<10,\,\,\,\,\, t>0\,</math><br>
*<math>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}\,</math><br><br>

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[[PDE12|solution]]<br><br>
<table border=0 cellspacing=0 width=100%><tr><td colspan=2 valign=top> Solve Dirichlet's problem for a circular annulus. The domain <math>D\,</math> is the space between two concentric circles, <math>C_1</math> being the innermost circle with radius <math>a</math>, and <math>C_2</math> being the outermost circle with radius <math>b</math>.</td></tr>
<tr><td width=30%>&nbsp;</td><td width=70%><math>\nabla^2 u = 0\,</math> in <math> D\,</math><br><math>u = g\,</math> on <math>C_1\,</math><br><math>u = f\,</math> on <math>C_2\,</math><br>
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</td></tr></table><br><br>
*<math>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]\,</math><br><br>


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