Math Help - integration general solution

1. integration general solution

Hi i have an integration problem. basically i am going through an example...the equation i want to find the gen solution is shown below..i have also shown the gen solution as well...but i cant understand how they got to that answers

$\xi_{yy}$ + $(1/y)\xi_{y}$=0

The general solution is given by

$\xi$ = $A(x)\ln(y) + B(x)$

can some one explain how they got to this general soution..thanks

2. Originally Posted by dopi
Hi i have an integration problem. basically i am going through an example...the equation i want to find the gen solution is shown below..i have also shown the gen solution as well...but i cant understand how they got to that answers

$\xi_{yy}$ + $(1/y)\xi_{y}$=0

The general solution is given by

$\xi$ = $A(x)\ln(y) + B(x)$

can some one explain how they got to this general soution..thanks
The function $\xi$ depends on $x$ and $y$ but the equation only involves $y$, hence it can be seen as an ordinary differential equation.

Let $x$ be fixed. Write $f(y)=\xi(x,y)$. Then $f$ is a function of 1 variable and $f''(y)+\frac{1}{y}f'(y)=0$ for every $y$. Let $g=f'$, so that $g'(y)+\frac{1}{y}g(y)=0$. This is a linear first order differential equation: we know how to solve it. Since $\int \frac{dy}{y}=\ln y$, there exists $A\in\mathbb{R}$ such that $g(y)=A e^{-\ln y}=\frac{A}{y}$. Hence, by integration, $f(y)=A\ln y + B$ for some $B\in\mathbb{R}$.
However, the "constants" $A$ and $B$ depend on $x$ which was fixed before, hence we should write $A(x)$ and $B(x)$: $\xi(x,y)=f(y)=A(x)\ln y+B(x)$. This is it.