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

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- Nov 9th 2013, 04:53 AMSuvadipfirst 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 AMHallsofIvyRe: 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 AMSuvadipRe: 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 AMHallsofIvyRe: 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 PMSuvadipRe: 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 PMProve ItRe: 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 PMSuvadipRe: first order ordinary differential equation
So the two solutions are $\displaystyle 5y^{3/5}=180x (\mbox{when} \quad y \neq 0)\quad \mbox{and}\quad y=0$?

- Nov 9th 2013, 06:48 PMProve ItRe: first order ordinary differential equation
Yes