Find the particular solution to the nonhomogeneous equation that does not involv

Question

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) =

Explanation / Answer

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

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