On x,y>0, transform y²u_xx−x²u_yy+yu_y=0 using η=x²/2, ξ=y²/2. It becomes $(u_{\eta\eta}-u_{\xi\xi})+(u_\eta+u_\xi)/(2\eta)+(a u_\eta+b u_\xi)/(2\xi)=0$. Find (a,b).
Chain rule gives u_xx=u_η+x²u_ηη and u_yy=u_ξ+y²u_ξξ, with yu_y=y²u_ξ. Divide the transformed equation by x²y²: $u_{\eta\eta}-u_{\xi\xi}+u_\eta/(2\eta)+u_\xi/(2\eta)-u_\xi/(2\xi)=0$. Compare coefficients: **a=0,b=−1, B**.
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