[SPLIT] 4y(x) + y''(x) = x has complex characteristic

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4y(x) + y''(x) = x

This one is very similar to my previous equation, but it has complex numbers in characteristic equation \(\displaystyle \L\\r^{2}+4=0\). How do I calculate them? :oops:
 
Use sine and cosine when they're complex.

You should be able to solve the quadratic.

\(\displaystyle \L\\r^{2}=-4\)

\(\displaystyle \L\\r=\sqrt{-4}\)

\(\displaystyle \L\\r = 2i \;\ and \;\ -2i\)

The solution is:

\(\displaystyle \L\\y=C_{1}cos(2x)+C_{2}sin(2x)+\frac{x}{4}\)

Now, it's your mission to get there. Okey-doke?.
 
If your ODE is

y" + By' + C = 0

Then your CE

r^2 + Br + C = 0

whose solutins would be

r_1 & r_2 = M ± i N

Then the homogeneous solution of the ODE is:

y = P * e^(r_1) + Q * e^(r_2)[/tex]
 
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