Question

Classify and reduce the partial differential equation to it's canonical form

Uxx+U2xy+Uyy =0

Answer 1

Given PDE is-

$U_{xx}+2U_{xy}+U_{yy}=0$

or, $\dfrac{d^2U}{dx^2}+\dfrac{2d^2U}{dxdy}+\dfrac{d^2U}{dy^2}=0$

i.e $. r+2s+t=0$

Then The required canonical form is-

$rR+sS+tT+\lambda(x,y,z,p,q)=0$

Where $R=1,S=2,T=1$

$S^2-4RT=(2)^2-4(1)(1)=0$

Given Equation is parabolic.

Also, $R\lambda^2+S\lambda+T=0$

$\lambda^2+2\lambda+1=0\\(\lambda+1)^2=0\\\lambda=-1,-1$

Then the characterstics equation is-

$\dfrac{dy}{dx}+\lambda=0\\ \dfrac{dy}{dx}-1=0\\dy-dx=0$

Integrating both the sides and we get the required canonical form as-

$y-x=c$