Homework excercise: Seperable ODEs and implicit equations

Valentin

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Sep 19, 2018
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Hello

I am stuck on a problem with the following decription:

Skærmbillede 2018-09-19 kl. 22.47.03.jpg

I do not exactly know what it means to show that the implicit solutions are determined via the ODE, and I cannot seem to show that the "!"-marked identities are true. I have tried two things:

(1) I have tried to establish each of the "!"-marked identities in the differential equation by computing the fraction with the partial derivatives:
Skærmbillede 2018-09-19 kl. 22.50.35.jpg

(2) I have tried to use the folowing theorem to determine the solutions to the differential equation:

Skærmbillede 2018-09-19 kl. 22.43.58.jpg

The theorem gives me the desired implicit equation, but with an integration constant added to it:

Skærmbillede 2018-09-19 kl. 22.52.17.jpg

I feel like this gets me close to solving the exercise. However I am completely disregarding the integration limits in the theorem (Since my problem is not an initial-value-problem I think it makes sense, but I am not sure). Even if I am close, I still need to show that the "!"-marked identities are true.

Some inputs on this would be greatly appreciated!
 

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