d^2y/dx^2+4dy/dx+13y=2cos2x
find the complementary function and particular integral.Hence write down the full general solutions.

Question

d^2y/dx^2+4dy/dx+13y=2cos2x
find the complementary function and particular integral.Hence write down the full general solutions.

zepdrix · Answer

eyyyy :)
What part are you having trouble with?

zepdrix · Answer

If we look for just the solutions to the homogeneous first,$$\large\rm y''+4y'+13y=0$$We get a characteristic equation that looks like this,$$\large\rm r^2+4r+13=0$$Understand that part? :)

anonymous · Answer

ys i know 
but how to solve when right having 2cos2x

zepdrix · Answer

Were you able to find the homogeneous solution?
It should give you something like this:$$\large\rm y_h=c_1e^{-2x}\cos(3x)+c_2e^{-2x}\sin(3x)$$Then to find the particular solution,
we assume that it has the form of the right side,
more generally though, sines and cosines.
$$\large\rm y_p=A\sin(2x)+B\cos(2x)$$

zepdrix · Answer

And then we take a couple derivatives to find our $\large\rm y'_p$ and $\large\rm y''_p$ and then plug all of that back into the orginal differential equation.

zepdrix · Answer

What do you think? :d
Too confusing?

anonymous · Answer

oh i knew now! thx

zepdrix · Answer

cool! :)