Find the solution of differential equation that passes through the indicating points
$dy/dx-y^2 = -9$ (0,3)
I have tried to solve it
$dy/dx = -9 + y^2$
$dy/dx = (y)^2 – (3)^2$
$dy/dx= (y-3).(y+3)$
$dy= (y-3).(y+3) dx$
What next ?
[Math] find the solution of differential equation that passes through the indicating points
ordinary differential equations
Best Answer
There is a simple way of solving any equation of the type $y'= f(x)g(y)$. There is some cheating in the notation in the derivation that follows, however it is the quickest way to memorize it, so as long as you know where you are going, it is acceptable to do the steps outlined.
$$\frac{dy}{dx} = f(x)\cdot g(y)\text{ ("multiply" by }dx, \text{ divide with }g(y))\\ \frac{dy}{g(y)}= \frac{dx}{f(x)} \text{ (integrate)}\\ \int\frac{dy}{g(y)} = \int\frac{dx}{f(x)} + C $$ This way, you get two functions, $F$ and $G$, and an equation $G(y) = F(x) + C$. Solving it for $y$ gives you the general form for the solution.