(See Steps) Using v=ln u and v=F(x)+g(y), show that the solution of the Cauchy problem y^2u_x^2+x^2u_y^2=(xyu)^2, u(x,0)=\exp (x^2) is u(x,y)=\exp
Question: Using \(v=\ln u\) and \(v=F\left( x \right)+g\left( y \right)\), show that the solution of the Cauchy problem
\[{{y}^{2}}{{u}_{x}}^{2}+{{x}^{2}}{{u}_{y}}^{2}={{\left( xyu \right)}^{2}},\,\,u\left( x,0 \right)=\exp \left( {{x}^{2}} \right)\]is
\[u\left( x,y \right)=\exp \left( {{x}^{2}}+i\frac{\sqrt{3}}{2}{{y}^{2}} \right)\]
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