I have stucked in heat Equation $u_{t}=u_{xx}$
I look for a solution of type $v(\frac{x^{2}}{t})$.
I set $z=\frac{x^{2}}{t}$ and i end up in the ode: $\frac{16}{t}v''(z)+(\frac{8}{x^{2}}+\frac{4t}{x^{3}}-1)v'(z)=0$
Because,
$v_{t}(x,t)=v'(z)\frac{x^{4}}{t^{4}}$
$u_{xx}(x,t)=v''(z)\frac{16x^{4}}{t^{4}}+v'(z)\frac{8x^{2}}{t^{2}}+v'(z)\frac{4x}{t^{2}}$
According to Lawrence C Evans in Partial Differential Equation in page 87,we should end up in the ode: $4zv''(z)+(2+z)v'(z)=0$
Who i am wrong?