The setting is the following: we have and if we reduce this to a first order pde by substituting , we get the following equations: . My question is why is there a and a after the substitution? Should it not be just and without a dot? Thank you.
The setting is the following: we have and if we reduce this to a first order pde by substituting , we get the following equations: . My question is why is there a and a after the substitution? Should it not be just and without a dot? Thank you.
I have no idea why those substitutions are necessary, but this question is just screaming for separation of variables.
Let $\displaystyle \begin{align*} \phi = X\left( x \right)Y\left( y \right) \end{align*}$ which gives
$\displaystyle \begin{align*} \frac{\partial \phi}{\partial x } &= X'\left( x \right) Y\left( y \right) \\ \frac{\partial ^2 \phi}{\partial x^2} &= X''\left( x \right) Y\left( y \right) \\ \\ \frac{\partial \phi}{\partial y} &= X\left( x \right) Y'\left( y \right) \\ \frac{\partial ^2 \phi}{\partial y^2 } &= X\left( x \right) Y''\left( y \right) \end{align*}$
so substituting into the DE gives
$\displaystyle \begin{align*} X\left( x \right) Y'' \left( y \right) - X''\left( x \right) Y \left( y \right) &= 0 \\ X\left( x \right) Y''\left( y \right) &= X''\left( x \right) Y\left( y \right) \\ \frac{ Y''\left( y \right) }{Y\left( y \right) } &= \frac{X''\left( x \right) } {X\left( x \right) } \end{align*}$
The only way for the LHS and RHS to be equal is if they both equal the same constant value (k) so by doing that you get two second order constant coefficient DEs you can solve for x and y.