integrating factor ODE particular solution

[ferrika3]

Hello again, I have completed an exercise but I am not really sure if the solution is the correct one... could someone help me out?
I was able to do a) the confirmation that I need is if the general solution for b) and the particular solution for c) are correct.



or should I proceed in this way and obtain 4 Cs?

 

[nasi112]

Your final solution should be.

[MATH]y = \frac{x^4(28y^{3/4} - 1)}{128} - \frac{7x^2y^{3/4}}{2}[/MATH]

 

[ferrika3]

Thank you for the reply, but I am not really sure of how you got to this answer, why is y also to the right?

I am quite sure of the general solution of B) as I checked with different online calculators and re did all the calculations, what I am most unsure about is for part C) how to obtain the constant C value, since if I were to expand the general solution equation and then resolve it I would get 4 different constant values, would that be normal?

 

[Deleted member 4993]

Thank you for the reply, but I am not really sure of how you got to this answer, why is y also to the right?

I am quite sure of the general solution of B) as I checked with different online calculators and re did all the calculations, what I am most unsure about is for part C) how to obtain the constant C value, since if I were to expand the general solution equation and then resolve it I would get 4 different constant values, would that be normal?

you say:

I am quite sure of the general solution of B) as I checked with different online calculators and re did all the calculations,

Have you checked whether "your solution" satisfies the original DE?

 

[Deleted member 4993]

Your final solution should be.

[MATH]y = \frac{x^4(28y^{3/4} - 1)}{128} - \frac{7x^2y^{3/4}}{2}[/MATH]

Have you checked whether "your solution" satisfies the original DE?

 

[HallsofIvy]

Your differential equation is \(\displaystyle \frac{du}{dx}- \frac{u}{x}= \frac{7}{4}x\).

An "integrating factor" is a function, \(\displaystyle \phi(u, x)\) such that
\(\displaystyle \frac{d(\phi u )}{dx}= \phi \frac{du}{dx}+ \frac{d\phi}{dx}u- \frac{\phi u}{x}\),

 

[nasi112]

Have you checked whether "your solution" satisfies the original DE?

I think that I have made a mistake.

Thank you for the reply, but I am not really sure of how you got to this answer, why is y also to the right?
I am quite sure of the general solution of B) as I checked with different online calculators and re did all the calculations, what I am most unsure about is for part C) how to obtain the constant C value, since if I were to expand the general solution equation and then resolve it I would get 4 different constant values, would that be normal?

I think that the steps you have used to solve is correct, but it is difficult to keep truck of of this monster DE. It needs some patient and I am not patient. It would be also good to follow what HallsofIly have said.


The main idea is that you have a Bernoulli differential equation. When you substitute [MATH]y = u^4[/MATH], you will get a linear differential equation that can be solved easily. Other things is just a matter of algebra and simplification.