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Supposedly simple differential equation

  • Thread starter popo902
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  • #1
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Homework Statement


Find the solution of
dy/dt = 1/ (e^y -t), y(1) = 0


Homework Equations





The Attempt at a Solution


i tried separating the equation, but the subtraction gets in the way
well this is what i have
y = t - 1 + C/e^t
i solved for t then i put that into the single order ODE formula
but the answer is y = arccosht
...so...
can someone give me a hint on how to start this?
 

Answers and Replies

  • #2
HallsofIvy
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Yes, the fact that an equation is not separable does make it hard to separate!

Here's what I would try. Since it is the [itex]e^y- t[/itex] that is the problem, let [itex]u= e^y- t[/itex] so that
[tex]\frac{dy}{dt}= \frac{1}{u}[/tex]
and since [itex]e^y= u+ t[/itex]
[tex]e^y\frac{dy}{dt}= \frac{du}{dt}+ 1[/tex]
[tex](u+ t)\frac{1}{u}= \frac{du}{dt}+ 1[/tex]
[tex]\frac{du}{dt}= \frac{u+t}{u}- 1= \frac{t}{u}[/tex]
which is separable.
 
  • #3
dextercioby
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Well, you can switch the variables and try to find t(y). The ODE which results

[tex] \frac{dt(y)}{dy} + t(y) = e^y [/tex]

should be very easy to integrate, right ?
 
  • #4
HallsofIvy
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Well, if you want to do it the easy way! :tongue2:
 

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