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How do you solve the fallowing ode?

u'=u^2

What I did was:

$ \frac{du}{dt}=u^2 \rightarrow du=u^2dt\rightarrow\int du=\int u^2dt\rightarrow u=u^2(t+c)\rightarrow u=\frac{1}{t+c} $

but the correct answer is $ u=\frac{1}{-t+c} $ Where was I wrong?

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    $u=0$ is also a solution, but it's not an interesting solution.2012-02-21

2 Answers 2

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Here is the correct way to do it:

First, divide by $u^2$ to get

u'/u^2=1.

Then integrate both sides to get

$-1/u=t+d$.

Finally, rearranging and letting $c=-d$ yields

$u=\frac{1}{c-t}$

as desired.

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    As Ilya mentioned $u=0$ is also a solution.2012-02-21
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Use the method of separation of variables:

$\frac{du}{dt}=u^2\Longrightarrow u^{-2}du=dt,$ $\int u^{-2}du=\int dt\Longrightarrow -u^{-1}=t+c_1\Longrightarrow u=\frac{1}{c_1-t}.$

Remember that $c_1$ can 'absorb' the negative sign.