Find a particular solution to y16y32sec4tSolutiony16y32sec4t

Find a particular solution to y??+16y=?32sec(4t).

Solution

y\"+16y=-32sec(4t)

Auxiliary equation D^2+16=0

D= +4i, -4i

Complementary function

Yc = C1 cos(4t) + C2 sin(4t)

Y1 = cos(4t) , Y2= sin(4t)

W= 4

Particular Integral:

Yp= -Y1 Integral Y2 T/W dt + Y2 Integral Y1 T/W dt

Where T= -32 sec(4t)

Yp= - cos(4t) Integral sin(4t) (-32)sec(4t)/4 dt +

sin(4t) Integral cos(4t) (-32)sec(4t)/4 dt

Yp= 8 cos(4t) Integral sin(4t) sec(4t) dt - 8 sin(4t) Integral

( cos(4t) sec(4t) dt )

Yp= -8t sin(4t) - cos(4t) - cos(4t) ln(|sin^2(4t) -1|)

Y= Yc+Yp

Y= C1 cos(4t) + C2 sin(4t) -8t sin(4t) - cos(4t) -

cos(4t) ln(|sin^2(4t-1|)