# Thread: first order ordinary differential equation

1. ## first order ordinary differential equation

How many solutions does the differential equation $\frac{dy}{dx}=60(y^2)^{1/5}; \quad x>0, y(0)=0$ have?

2. ## Re: first order ordinary differential equation

First, $(y^2)^{1/5}$ is a very strange way to write $y^{2/5}$. Is that what you meant?

If it is, then this is a separable equation that is easily solvable: $\dfrac{dy}{y^{2/5}}= y^{-2/5}dy= 60 dx$.

What do you get when you integrate both sides?

3. ## Re: first order ordinary differential equation

Ya this is easily solvable. But how many solutions does it have? The answer was given as 2. But I cant understand how? It was set in an entrance exam.

4. ## Re: first order ordinary differential equation

Start by solving it! That should help you see what solutions there are!

Have you considered what would happen if y were identically 0?

5. ## Re: first order ordinary differential equation

I have arrived at the solution $5y^{3/5}=3x$. This is the only solution I got. Then how the no of solution is 2?

6. ## Re: first order ordinary differential equation

Did you even read HallsofIvy's last post? You can only solve the equation through dividing, which means you're making the assumption that \displaystyle \begin{align*} y \neq 0 \end{align*}. WHAT IF IT WAS?!?!

7. ## Re: first order ordinary differential equation

So the two solutions are $5y^{3/5}=180x (\mbox{when} \quad y \neq 0)\quad \mbox{and}\quad y=0$?

Yes