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Given the differential equation $$ \frac{dy}{dx} = y+x$$

I am told this differential equation is separable. Meaning I need to rewrite the RHS into a product of two variables depending on y and x.

I've tried for some time now but I simply cannot figure out how this is separable. I'm able to solve it using the method of "integrating factor" and so I know the solution should be $$ y = Ce^x-x-1 $$

Any ideas?

Raffaele
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Sirmimer
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  • It's clearly not separable. Possibly someone lied, possibly you misunderstood (maybe it was said that the DE can be converted to a separable equation or somethinglike that?) – David C. Ullrich Feb 07 '21 at 13:57
  • It states: The equation is separable. Solve it using the learnt method (where you separate the variables). But I'm thinking there might be an error in the textbook (and I'm suppose to just use the integration factor method) – Sirmimer Feb 07 '21 at 14:49

1 Answers1

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Perhaps with $z:=y+x+1$, $$ \frac{\mathrm dz}{\mathrm dx}= \frac{\mathrm dy}{\mathrm dx}+1=y+x+1=z.$$

Hagen von Eitzen
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  • I played around with it a little bit, but I couldn't get it working. I end up with $$dz=dy+dx$$. Then I want to expand $$z = y + x + 1$$ and end up with something wrong – Sirmimer Feb 07 '21 at 12:50
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    I think the $+1$ is a bit artificial, proceeding with $z=x+y$ and ODE gives $\frac{dy}{dx}=z\implies dz=dx+dy=dx+zdx=(z+1)dx$ feels more intuitive. – zwim Feb 07 '21 at 13:29
  • I found this thread: https://math.stackexchange.com/questions/537629/is-the-differential-equation-y-xy-separable

    Here it states that is it NOT possible

     – Sirmimer Feb 07 '21 at 14:48
  • It is not separable in $x,y$ but it is after a change of variables (namely it is separable in $x,z$). – zwim Feb 07 '21 at 17:47