v = \(\frac{c}{\sqrt t}e^{\frac{-x^2}{4a^2t}}\)
\(\frac{\partial v}{\partial x}=\frac{-2x}{4a^2t}\frac{c}{\sqrt t}e^{\frac{-x^2}{4a^2t}}\)
\(=\frac{-2x}{4a^2t}v\)
\(\frac{\partial^2v}{\partial x^2}=\frac{-2v}{4a^2t}-\frac{-2x}{4a^2t}.\frac{\partial v}{\partial x}\)
\(=\frac{-2v}{4a^2t}+\frac{2x}{4a^2t}.\frac{2x}{4a^2t}v\)
\(=\frac{-2v}{4a^2t}(1-\frac{2x^2}{4a^2t})\)
\(a^2\frac{d^2v}{dx^2}=\frac{-v}{2t}(1-\frac{x^2}{2a^2t})\)-----(i)
\(\frac{\partial v}{\partial t}=\frac{c}{\sqrt t}e^{-\frac{x^2}{4a^2t}}\) x \(\frac{x^2}{4a^2t^2}-\frac{c}{2t^{3/2}}e^{-\frac{x^2}{4a^2t}}\)
\(=\frac{x^2v}{4a^2t^2}-\frac{v}{2t}\)
\(= -\frac{v}{2t}(1-\frac{x^2}{2a^2t})\)
\(=a^2\frac{\partial^2v}{\partial x^2}\) (From (i))
