Is this the right solution for the ODE

[Javierlgc]

Homework Statement

well the problem is to solve de following differential equation.

$y'^3+(x+2)e^y=0$

Homework Equations

$y'=dy/dx=p$

The Attempt at a Solution

I got this problem in my test today, an i did it just like it is in the image below, but my teacher wasn't sure that it was a correct way of solving it, i would like to know if it is, and if it's not and how to solve it them.

I got this from a book of solutions of de T.Mackarenko, and I think is right but i know that the constant in the end when you have integrated has to show the highest power of $p$ in this case 3, but also i don't know if in this case it changes because i made it a different equation in which $p$ was to the power of 1. Thank you

 

[HallsofIvy]

I don't know what you mean by " the constant in the end when you have integrated has to show the highest power of p in this case 3". The constant of integration is just that- a constant- a number. It doesn't "show a power"

 

[Javierlgc]

I don't know what you mean by " the constant in the end when you have integrated has to show the highest power of p in this case 3". The constant of integration is just that- a constant- a number. It doesn't "show a power"

Well I've learned that the constant of integration at the end of a differential equation is going to show you the power to which the derivative was. Is like in algebraic equations when you factorize a polynomial of the 5th power when you take the factorization back out you will get again a polynomial of the 5th power. The same i was toughed with differential equations, if your ODE power is 3, you will end up with a constant to the power of 3. Sorry if my english is not clear.

But, is the solution correct?

 

[Redbelly98]

Welcome to Physics Forums!

Well I've learned that the constant of integration at the end of a differential equation is going to show you the power to which the derivative was. Is like in algebraic equations when you factorize a polynomial of the 5th power when you take the factorization back out you will get again a polynomial of the 5th power. The same i was toughed with differential equations, if your ODE power is 3, you will end up with a constant to the power of 3. Sorry if my english is not clear.

I'm unfamiliar with that technique, but I am a physicist and not a mathematician.

But, is the solution correct?

To check, you can differentiate your final expression, and see if you can work it to get the original differential equation. That being said, yes, it looks correct. But maybe not in final acceptable form? It may be necessary to solve explicitly for y to earn full credit.

Separation of variables is a pretty standard, elementary technique. I'm a little surprised that your teacher seems to be unfamiliar with it.