Question #185340

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

Uxx+U2xy+Uyy =0


1
Expert's answer
2021-05-07T10:00:58-0400

Given PDE is-

Uxx+2Uxy+Uyy=0U_{xx}+2U_{xy}+U_{yy}=0


or, d2Udx2+2d2Udxdy+d2Udy2=0\dfrac{d^2U}{dx^2}+\dfrac{2d^2U}{dxdy}+\dfrac{d^2U}{dy^2}=0


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


Then The required canonical form is-

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


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


S24RT=(2)24(1)(1)=0S^2-4RT=(2)^2-4(1)(1)=0

Given Equation is parabolic.


Also, Rλ2+Sλ+T=0R\lambda^2+S\lambda+T=0

λ2+2λ+1=0(λ+1)2=0λ=1,1\lambda^2+2\lambda+1=0\\(\lambda+1)^2=0\\\lambda=-1,-1


Then the characterstics equation is-

dydx+λ=0dydx1=0dydx=0\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-

yx=cy-x=c


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