UC

LI = { t^2 Sin[t] , t Sin[t] , t^2 Cos[t] , t Cos[t],Sin[t],Cos[t]}

yp[t_] = a1 t^2 Sin[t] + a0 t Sin[t] + b1 t^2 Cos[t] + b0 t Cos[t]

yp'[t]

Collect[%,LI]

yp''[t]

Collect[%,LI]

yp'''[t]

Collect[%,LI]

yp'''[t] + yp'[t]

Collect[%,LI]

Solve[{(6 a1-2 b0) ==0,-4 b1 ==0,+(-2 a0-6 b1) ==0,-4 a1 ==1}]

( a1 t^2 Sin[t] + a0 t Sin[t] + b1 t^2 Cos[t] + b0 t Cos[t])/.{a1->-(1/4),b0->-(3/4),b1->0,a0->0}

VP

D[ u1[t] + u2[t] Sin[t] + u3[t] Cos[t],t]

eq1 = (u1'[t]+Sin[t] u2'[t]+Cos[t] u3'[t]==0);

D[Cos[t] u2[t]-Sin[t] u3[t],t]

eq2 = (Cos[t] u2'[t]-Sin[t] u3'[t]==0);

D[-Sin[t] u2[t]-Cos[t] u3[t],t]

(-Cos[t] u2[t]+Sin[t] u3[t]-Sin[t] u2'[t]-Cos[t] u3'[t]) + (Cos[t] u2[t]-Sin[t] u3[t])

RowReduce[{{1,Sin[t],Cos[t],0},{0,Cos[t],-Sin[t],0},{0,-Sin[t],-Cos[t],t Sin[t]}}]

Integrate[{t Sin[t],-t ((Sin[t])^2),- t Cos[t] Sin[t]},t]

%.{1,Sin[t],Cos[t]}

Integrate[t Sin[t],t]

Integrate[-t ((Sin[t])^2),t]

Integrate[-t Sin[t] Cos[t],t]

Expand[(-t Cos[t]+Sin[t]) + (-((t^2)/4)+(1/8) Cos[2 t]+(1/4) t Sin[2 t])*Sin[t] + ((1/4) t Cos[2 t]-(1/8) Sin[2 t])*Cos[t]]

TrigExpand[%]

Expand[{-t Cos[t]+Sin[t],(-((t^2)/4)+(1/8) (2 (Cos[t])^2 - 1)+(1/4) t (2 Sin[t] Cos[t]))*Sin[t] ,((1/4) t (1-2(Sin[t])^2)-(1/8) (2 Sin[t] Cos[t]))*Cos[t]}]

Expand[(-t Cos[t]+Sin[t]) + (-((t^2)/4)+(1/8) (2 (Cos[t])^2 - 1)+(1/4) t (2 Sin[t] Cos[t]))*Sin[t] + ((1/4) t (1-2(Sin[t])^2)-(1/8) (2 Sin[t] Cos[t]))*Cos[t]]

reconcile

Plot[-(3/4) t Cos[t]-(1/4) (t^2) Sin[t],{t,-2Pi,2Pi}]

Plot[-(3/4) t Cos[t]+((7 Sin[t])/8)-(1/4) (t^2) Sin[t],{t,-2Pi,2Pi}]

ICs

ygen[t_] = -(3/4) t Cos[t]-(1/4) (t^2) Sin[t] + c1 + c2 Sin[t] + c3 Cos[t]

ygen'[t]

ygen''[t]

ygen[0]

ygen'[0]

ygen''[0]

Solve[{c1+c3==3,-(3/4)+c2==1,-c3==-2}]