Given PDE is-
Uxx+2Uxy+Uyy=0
or, dx2d2U+dxdy2d2U+dy2d2U=0
i.e .r+2s+t=0
Then The required canonical form is-
rR+sS+tT+λ(x,y,z,p,q)=0
Where R=1,S=2,T=1
S2−4RT=(2)2−4(1)(1)=0
Given Equation is parabolic.
Also, Rλ2+Sλ+T=0
λ2+2λ+1=0(λ+1)2=0λ=−1,−1
Then the characterstics equation is-
dxdy+λ=0dxdy−1=0dy−dx=0
Integrating both the sides and we get the required canonical form as-
y−x=c
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