Is this the right solution for the ODE

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In summary, the homework statement is trying to solve a differential equation, but is unsure of how to do it.
  • #1
Javierlgc
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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.

347wths.jpg


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
 
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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"
 
  • #3
HallsofIvy said:
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?
 
  • #4
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Javierlgc said:
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.
 

FAQ: Is this the right solution for the ODE

Is this the right solution for the ODE?

The answer to this question depends on what the ODE is and what specific solution you are referring to. Generally, to determine if a solution is right for an ODE, you can plug the solution into the ODE and see if it satisfies the equation. If it does, then it is a valid solution. Additionally, you can check if the solution satisfies any initial or boundary conditions that may be given.

How do I know if my solution is correct?

To determine if your solution is correct, you can follow the same steps as mentioned above. Plug the solution into the ODE and see if it satisfies the equation and any given initial or boundary conditions. You can also double check your work by taking the derivative of the solution and seeing if it matches the original equation.

Can I use a different method to solve this ODE?

Yes, there are multiple methods for solving ODEs, such as separation of variables, substitution, and power series. The method you choose may depend on the specific ODE and your personal preference. It is always a good idea to try different methods and compare the solutions to ensure accuracy.

How do I check the accuracy of my solution?

One way to check the accuracy of your solution is to compare it to a known solution, if one is available. You can also use numerical methods, such as Euler's method or the Runge-Kutta method, to approximate the solution and compare it to your analytical solution. Additionally, you can plot the solution and see if it matches the behavior of the ODE.

Can I use this solution for all values of the independent variable?

This depends on the specific solution and ODE. Some solutions may only be valid for certain values of the independent variable, while others may be valid for all values. It is important to check the domain of the solution and see if it aligns with the domain of the ODE. If there are any restrictions, they should be noted and taken into account when using the solution.

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