Y"+4y = 8 sin 2x. Trial Solution using annihilator technique Linear Algebra and Differential equations I am so confused why would you multiple by X up(x)

Question

asked May 16, 2020 5.3k views

1 Answer

Vasilis Vasilatos · Answer 1 · 2020-05-21T10:05:18+0000

A second order linear, non - homogeneous ODE has a form of
ay''+by'+cy=g(x)

The general solution to,
a(x)y''+b(x)y'+c(x)y=0

Can be written as,

y=y_h+y_p

Where
y_h is a solution to the homogeneous ODE and
y_p the particular solution, function that satisfies the non - homogeneous equation.

We can solve
y_h by rewriting the equation,

$ay''+by'+cy=0\Longrightarrow(e^(xy))''+4e^(xy)=0$

Which simplifies to,

e^(xy)(y^2+4)=0

From here we get two solutions,

y_(h1)=2i, y_(h2)=-2i

So the form here refines,

$y_h=c_1\cos(2x)+c_2\sin(2x)$

The same thing we do with
y_p to get form of,

$y_p=-2x\cos(2x)$

From here the final form emerges,

$y=\boxed{c_1\cos(2x)+c_2\sin(2x)-2x\cos(2x)}$

Hope this helps.

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