Without solving for the undetermined coefficients the correc

Without solving for the undetermined coefficients, the correct form of a particular solution of the differential equation y\" + 6y\' + 13y = e^-3x cos(2x) is

Solution

Given that

y\'\' + 6y\' + 13y = e^-3xcos(2x)

D-operator form is ,

( D² + 6D + 13 ) y= e^-3xcos(2x)

Auxialary equation is ,

m² + 6m + 13 = 0

On solving ,

m = -1 ± 23 i

Complementary function y_c = e^-x [ c₁cos (23)x + c₂sin (23)x ]

The right hand side of the given equation is a product of an exponential and cosine function.

Hence,

Perticular solution = e^-3x ( A sin(2x) + B cos(2x) )