1. ## dy/dx = -y/x

Hello everybody,

I need to solve

dy/dx = -y/x

for some economics problem. Can anybody help me?

Thanks,

Rob

2. The DE you propose can be written as...

$\displaystyle \frac {dy}{y} = -\frac{dx}{x}$ (1)

Al thet you have to do is to integrate first and second term of (1) indicating the 'arbitrary constant' as $\displaystyle \ln c$...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$

3. Originally Posted by robvaneijk
Hello everybody,

I need to solve

dy/dx = -y/x

for some economics problem. Can anybody help me?

Thanks,

Rob
$\displaystyle \frac{dy}{dx} = -\frac{y}{x}$

$\displaystyle \frac{1}{y}\,\frac{dy}{dx} = -\frac{1}{x}$

$\displaystyle \int{\frac{1}{y}\,\frac{dy}{dx}\,dx} = \int{-\frac{1}{x}\,dx}$

$\displaystyle \int{\frac{1}{y}\,dy} = -\int{\frac{1}{x}\,dx}$

$\displaystyle \ln{|y|} + C_1 = -\ln{|x|} + C_2$

$\displaystyle \ln{|y|} + \ln{|x|} = C$, where $\displaystyle C = C_2 - C_1$

$\displaystyle \ln{|y||x|} = C$

$\displaystyle \ln{|xy|} = C$

$\displaystyle |xy| = e^C$

$\displaystyle xy = \pm e^C$

$\displaystyle xy = A$ where $\displaystyle A = \pm e^C$

$\displaystyle y = \frac{A}{x}$.