Solve the following differential equations.
y' = ((y^2)+xy)/(x^2)
Whats the method of solving this? What theorem am i using?
First divide by $\displaystyle y^2$ both sides : $\displaystyle
\tfrac{{y'}}
{{y^2 }} = \tfrac{1}
{{x^2 }} + \tfrac{1}
{x} \cdot \tfrac{1}
{y}
$
But note that: $\displaystyle
\tfrac{{y'}}
{{y^2 }} = - \left( {\tfrac{1}
{y}} \right)^\prime
$
Let: $\displaystyle
\tfrac{1}
{y} = z
$ we have: $\displaystyle
- z' = \tfrac{1}
{{x^2 }} + \tfrac{1}
{x} \cdot z
$
You can solve this last one by the integrating factor method