Originally Posted by

**Danny** If $\displaystyle y X' Y' + X Y = 0$

then

$\displaystyle \frac{y Y'}{Y} = - \frac{X}{X'}$

Since the LHS is only a function of $\displaystyle y$ and the RHS only a function of $\displaystyle x$ then each must be constant. Thus,

$\displaystyle \frac{y Y'}{Y} = - \frac{X}{X'} = \lambda$

1) $\displaystyle \frac{y Y'}{Y} = \lambda$. Separate $\displaystyle \frac{dY}{Y} = \frac{\lambda}{y}$ so $\displaystyle \ln Y = \lambda \ln y + \ln c_1$. Thus, $\displaystyle Y = c_1 y^{\lambda}$.

2) $\displaystyle - \frac{X}{X'} = \lambda$. Separate $\displaystyle \frac{dX}{X}= - \frac{dx}{\lambda}$ so $\displaystyle \ln X = - \frac{1}{\lambda} x + \ln c_2$. Thus, $\displaystyle X = c_2 e^{-x/\lambda}$.

Then multiply $\displaystyle X Y$ and combine your constants to a single constant.