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Here's the problem:

$\frac{dy}{dt}-y=7e^t + 25e^{6t}$ in terms of $\,y\,$, when $\,y(0)=7\,$

2 Answers 2

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To solve:

$y'-y=7e^t+25e^{6t}$

Multiply by $e^{-t}$:

$e^{-t}y'-e^{-t}y=7+25e^{5t}$

So that the right side is the result of the product rule, as follows:

$(ye^{-t})'=7+25e^{5t}$

Integrate both sides to find

$ye^{-t}=7t+5e^{5t}+C$ $y=7te^t+5e^{6t}+Ce^t$

Then just plug in the point $(t,y)=(0,7)$ given by your initial condition and solve for $C$. The resulting function for $y$ will be your answer. Though as a disclaimer I do have to point out that unless you have mathematically minded friends, this will probably impress them less than you'd like.

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    Mathematically minded friends wouldn't be impressed by this :P2012-08-31
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Put $y = a e^t + b te^t + c e^{6t}$ and solve for $a$, $b$, $c$.