## partial fractions (?) to solve first order DE

hello world,

I've been doing some summertime training to brush up my math skills and have been struggling with this

[dy]/[/dt]=(4exp(-y)+const*exp(-2y))^1/2

In fact this is the simplified version of a Bernouilli equation. I know that it is separable, I'm just struggling with the integration with respect to "y".

If anybody could help that'd be wonderful. Thanks a lot, have a nice day!

~huckleberry

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 Recognitions: Gold Member Science Advisor Staff Emeritus Your equation is $$\frac{dy}{dx}= \left(e^{-y}+ Ce^{-2y}\right)^{1/2}$$ which can be written as $$\frac{dy}{\left(e^{-y}+ Ce^{-2y}\right)^{1/2}}= dx$$ I would let $u= e^{-y}$ so that $du= -e^{-y}dy$ and $du/u= -dy$ In terms of u, the equation becomes $$\frac{du}{u\left(u+ Cu^2\right)^{1/2}}= dt$$
 HallsOfIvy, Thank you for you response. I tried your method and i've run up against a similar problem as before, when i tried partial fractions. I can't seem to solve correctly for the coefficients. A/u +B/(u+C*u^2)^(1/2)=1 A*(u+C*u^2)^(1/2) +Bu=1 choosing u=(-1/C) --> B=-C however, I get stuck looking A, the only way to make the B drop is to set u=0, which is not coherent. do you have a hint for me? thanks. ~huckleberry

## partial fractions (?) to solve first order DE

ps ( how do you enter the equations so nicely?)

 Recognitions: Homework Help Science Advisor I don't believe you can use partial fractions when there are surds involved. Try making a change of variable to get the surd in the form √(u2+B) (where B may be negative in this case), then look for a trig or hyperbolic trig substitution to make the surd collapse. To make your posts neater, click on "Go Advanced". That brings up a palette on the right from which you pick various symbols, and a toolbar above which makes e.g. superscript and subscript easy. To make it really pretty, click on the Ʃ symbol at the end of the toolbar. This brings up a Latex palette. You'll need to play around with that a bit to get the hang of it. If using either of these, remember to click Preview Post and check what it's going to look like before submitting.
 Recognitions: Gold Member Homework Help Science Advisor At some point, after completing the square and having eliminated the surd through haruspex's advice, you probably will need to make another substitution of te form v=Tan(u/2), or v=Tanh(u/2).