particular solutin, variation of constants

mathstresser

Junior Member
Joined
Jan 28, 2006
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134
Find a particular solution to

\(\displaystyle \L\\ x'' + x = (sect)^2\)

using the method of variation of constants.

I don't really know what to do with this.... This is all I can come up with.

Xp= u1x1 +u2x2

Wronskian=> W

W1/W, W2/W

\(\displaystyle \L\\ (sect)^2 = 1/(cost)^2\)
 
I believe that's called Variation of Parameters.

The auxiliary equation is \(\displaystyle \L\\m^{2}+1=0\)

So \(\displaystyle x_{c}=C_{1}cos(t)+C_{2}sin(t)\)

\(\displaystyle \L\\W=\begin{vmatrix}cos(t)&sin(t)\\-sin(t)&cos(t)\end{vmatrix}\)

Identify \(\displaystyle f(t)=sec^{2}(t)\)

\(\displaystyle \L\\u'_{1}=\frac{-sin(t)}{cos^{2}(t)}\)

\(\displaystyle \L\\u'_{2}=sec(t)\)

Then,

\(\displaystyle \L\\u_{1}=\frac{-1}{cos(t)}=-sec(t)\)

\(\displaystyle \L\\u_{2}=ln|sec(t)+tan(t)|\)

\(\displaystyle \L\\y=C_{1}cos(t)+C_{2}sin(t)-cos(t)sec(t)+sin(t)ln|sec(t)+tan(t)|\)

\(\displaystyle \L\\=C_{1}cos(t)+C_{2}sin(t)-1+sin(t)ln|sec(t)+tan(t)|\)

for \(\displaystyle \L\\\frac{-\pi}{2}<t<\frac{\pi}{2}\)

Does that look OK?. Check for errors.
 
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