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

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

The general solution is given by

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

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

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

The general solution is given by

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

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

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