Use the method of undetermined coefficients to find the gene

Use the method of undetermined coefficients to find the general solution of differential equation: y\" +2y\' + 5y - 3sin2t

Solution

Answer.

The auxiliary equation is:

r² + 2r + 5 = 0

r = -1 ± 2i

The homogeneous solution is:

y_h = c₁e^-x + c₂e^{± 2ix}

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Given g (x) = 3 sin (2t), the particular solution will be:

y_p = A sin (2t) + B cos (2t)

and its derivatives are:

y\'_p = A cos (2t) - B sin (2t)

y\'\'_p = -A sin (2t) - B cos (2t)

Replacing in given d.e:

y\'\' + 2y\' + 5y = 3 sin (2t)

-A sin (2t) - B cos (2t)

-2B sin (2t) + 2A cos (2t)

+5B sin (2t) +5A cos (2t)

When we take the summ of all of this:

(-A + 3B)sin (2t) + (7A - B)cos (2t) = 3 sin (2t)

-A + 3B = 3 ==> -1/3 A + B = 1

7A - B = 0   ==> 7 A     - B = 0

----

(-1/3+7)A = 1

A = 20/3

and B is;

7(20/3) - B = 0

B = 140/3

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Then the final particular solution is;

y_p = 20/3 sin (2t) + 140/3 cos (2t)

And the general solution is the sum of

y_h+ y_p = c₁e^-x + c₂e^{± 2ix} + 20/3 sin (2t) + 140/3 cos (2t)

and we are done!!