# first order ordinary differential equation

• Nov 9th 2013, 04:53 AM
first order ordinary differential equation
How many solutions does the differential equation $\displaystyle \frac{dy}{dx}=60(y^2)^{1/5}; \quad x>0, y(0)=0$ have?
• Nov 9th 2013, 07:48 AM
HallsofIvy
Re: first order ordinary differential equation
First, $\displaystyle (y^2)^{1/5}$ is a very strange way to write $\displaystyle y^{2/5}$. Is that what you meant?

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

What do you get when you integrate both sides?
• Nov 9th 2013, 08:32 AM
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.
• Nov 9th 2013, 09:04 AM
HallsofIvy
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?
• Nov 9th 2013, 05:59 PM
Re: first order ordinary differential equation
I have arrived at the solution $\displaystyle 5y^{3/5}=3x$. This is the only solution I got. Then how the no of solution is 2?
• Nov 9th 2013, 06:16 PM
Prove It
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 \displaystyle \begin{align*} y \neq 0 \end{align*}. WHAT IF IT WAS?!?!
• Nov 9th 2013, 06:34 PM
So the two solutions are $\displaystyle 5y^{3/5}=180x (\mbox{when} \quad y \neq 0)\quad \mbox{and}\quad y=0$?