PDE10

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u_{x}+2u_{y}=0,\,\,\,u(0,y)=3\,e^{{-2y}}\,

Make the seperation of variables u(x,y)=X(x)Y(y)\,.

Plug back into the DE.

X'Y+2XY'=0\,

{\frac  {X'}{X}}={\frac  {-2Y'}{Y}}=\lambda \,

This gives two ODES:

X'+\lambda X=0\,

X(x)=c_{1}\,e^{{\lambda x}}\,

-2Y'+\lambda Y=0\,

Y(y)=c_{2}\,e^{{{\frac  {\lambda }{2}}y}}\,

Let A=c_{1}c_{2}\,.

u(x,y)=A\,e^{{\lambda ({\frac  {y}{2}}-x)}}\,

Applying the boundary condition,

u(0,y)=A\,e^{{\lambda {\frac  {y}{2}}}}=3\,e^{{-2y}}\,

So A=3,\,\,\,\lambda =-4\,

The solution is

u(x,y)=3\,e^{{4(x-{\frac  {y}{2}})}}\,

Partial Differential Equations

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