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10.7.1: Variation of Parameters for Nonhomogeneous Linear Systems (Exercises)

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1 newcommand dy , mathrmdy newcommand dx , mathrmdx newcommand dyx , frac mathrmdy mathrmdx newcommand ds , mathrmds newcommand dt , mathrmdt newcommand\dst , frac mathrmds mathrmdt

Q10.7.1

In Exercises 10.7.1-10.7.10 find a particular solution.

1. y=[1411]y+[21e4t8e3t]

2. y=15[43211]y+[50e3t10e3t]

3. y=[1221]y+[1t]

4. y=[4365]y+[22et]

5. y=[6312]y+[4e3t4e5t]

6. y=[0110]y+[1t]

7. y=[311351624]y+[363]

8. y=[311232412]y+[1etet]

9. y=[322232223]y+[ete5tet]

10. y=13[113443210]y+[etetet]

Q10.7.2

In Exercises 10.7.11-10.7.20 find a particular solution, given that Y is a fundamental matrix for the complementary system.

11. y=1t[1tt1]y+t[costsint];Y=t[costsintsintcost]

12. y=1t[1tt1]y+[tt2];Y=t[etetetet]

13. y=1t21[t11t]y+t[11];Y=[t11t]

14. y=13[12et2et1]y+[e2te2t];Y=[2etet2]

15. y=12t4[3t3t613t3]y+1t[t21];Y=1t2[t3t41t]

16. y=[1t1ett1ett+11t+1]y+[t21t21];Y=t[tetett]

17. y=1t[110021222]y+[121];Y=[t2t31t22t3102t32]

18. y=[3ete2tet2ete2tet1]y+[e3t00];Y=[e5te2t0e4t0ete3t11]

19. y=1t[1t001t0t1]y+[ttt];Y=t[1costsint0sintcost0costsint]

20. y=1t[ett1etet1tetett1et]y+1t[et0et];Y=1t[etettetetetetet0]

Q10.7.3

21. Prove Theorem 10.7.1.

22.

  1. Convert the scalar equation P0(t)y(n)+P1(t)y(n1)++Pn(t)y=F(t) into an equivalent n×n system y=A(t)y+f(t).
  2. Suppose (A) is normal on an interval (a,b) and {y1,y2,,yn} is a fundamental set of solutions of P0(t)y(n)+P1(t)y(n1)++Pn(t)y=0 on (a,b). Find a corresponding fundamental matrix Y for y=A(t)y on (a,b) such that y=c1y1+c2y2++cnyn is a solution of (C) if and only if y=Yc with c=[c1c2cn] is a solution of (D).
  3. Let yp=u1y1+u1y2++unyn be a particular solution of (A), obtained by the method of variation of parameters for scalar equations as given in Section 9.4, and define u=[u1u2un]. Show that yp=Yu is a solution of (B).
  4. Let yp=Yu be a particular solution of (B), obtained by the method of variation of parameters for systems as given in this section. Show that yp=u1y1+u1y2++unyn is a solution of (A).

23. Suppose the n×n matrix function A and the n–vector function f are continuous on (a,b). Let t0 be in (a,b), let k be an arbitrary constant vector, and let Y be a fundamental matrix for the homogeneous system y=A(t)y. Use variation of parameters to show that the solution of the initial value problem

y=A(t)y+f(t),y(t0)=k

is

y(t)=Y(t)(Y1(t0)k+tt0Y1(s)f(s)ds).


This page titled 10.7.1: Variation of Parameters for Nonhomogeneous Linear Systems (Exercises) is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench.

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