I have a few questions about finding the general solution of first-order linear differential equations..
please help, I have no clue!! =/
http://i28.tinypic.com/2yv3343.gif
http://i31.tinypic.com/2818w02.gif
http://i28.tinypic.com/n81ee.gif
I have a few questions about finding the general solution of first-order linear differential equations..
please help, I have no clue!! =/
http://i28.tinypic.com/2yv3343.gif
http://i31.tinypic.com/2818w02.gif
http://i28.tinypic.com/n81ee.gif
it is just something you multiply through by to make the left hand side the derivative given by the product and the integrating factor. for the first one:
$\displaystyle y' + 2y = 3e^t$
(with practice, you will be able to recognize the integrating factor immediately, but let's go through the method to see it)
the integrating factor is $\displaystyle e^{\int 2~dt} = e^{2t}$
multiply through by the integrating factor, we get:
$\displaystyle e^{2t}y' + 2e^{2t}y = 3e^{3t}$
now the left hand side is the result of differentiating $\displaystyle e^{2t}y$ by the product rule, thus
$\displaystyle (e^{2t}y)' = 3e^{3t}$
integrate both sides, we get:
$\displaystyle e^{2t}y = e^{3t} + C$
$\displaystyle \Rightarrow y = e^t + Ce^{-2t}$
the others are done similarly