Find the particular solution to the nonhomogeneous equation

Find the particular solution to the nonhomogeneous equation that does not involve any terms from the homogeneous solution.

1 > 0 , has homogeneous solutions yi I)-. y2 1)-1 1 The nonhomogeneous equation t\'y -2y -8t1, t>0, has homogeneous solutions yIt)t >2lt)-Find the particular solution to the Zy = &2 1 Find the particular solution to the The nonhomogeneous equation t J nonhomogeneous equation that does not involve any terms from the homogeneous solution. Click here to enter or edit your answe y(t) =

Solution

Assume particular solution is:

yp=u y1+v y2

with the constraint

u\' y1+v\' y2=0

yp\' = u y1\'+v y2\'

yp\'\'=u\' y1\'+ v\' y2\' +u y1\'\' + v y2\'\'

Using the fact that y1 and y2 are solutions to homogeneous equation we get:

t^2(u\' y1\' +v\' y2\')=8t^2-1

t^2(2u\' t-v\'/t^2) =8t^2-1

Constraint is:

u\' y1+v\' y2=0

u\' t^2+v\'/t=0

v\'=-u\' t^3

Substituting gives

t^2 (3u\' t)=8t^2-1

3u\'=8/t-1/t^3

Integrating gives:

u=8/3 ln(t)+1/(6t^2)

3u\'=8/t-1/t^3 ,v\'=-u\' t^3

-u\'= -8/3 t+1/(3t^3)

v\'=-8 t^4/3+1/3

Integrating gives

v=-8t^5/15+t/3

So particular solution is

y1 u+ y2 v=

(8/3 ln(t)+1/(6t^2))t^2 +(-8t^5/15+t/3)/t

=8t^2/3 ln(t) +1-8t^4/15