10.7.1: Variation of Parameters for Nonhomogeneous Linear Systems (Exercises)
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In Exercises 10.7.1-10.7.10 find a particular solution.
1. y′=[−1−4−1−1]y+[21e4t8e−3t]
2. y′=15[−43−2−11]y+[50e3t10e−3t]
3. y′=[1221]y+[1t]
4. y′=[−4−365]y+[2−2et]
5. y′=[−6−31−2]y+[4e−3t4e−5t]
6. y′=[01−10]y+[1t]
7. y′=[31−1351−624]y+[363]
8. y′=[3−1−1−2324−1−2]y+[1etet]
9. y′=[−3222−3222−3]y+[ete−5tet]
10. y′=13[11−3−4−43−210]y+[etetet]
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[1t−t1]y+t[costsint];Y=t[costsint−sintcost]
12. y′=1t[1tt1]y+[tt2];Y=t[ete−tet−e−t]
13. y′=1t2−1[t−1−1t]y+t[1−1];Y=[t11t]
14. y′=13[1−2e−t2et−1]y+[e2te−2t];Y=[2e−tet2]
15. y′=12t4[3t3t61−3t3]y+1t[t21];Y=1t2[t3t4−1t]
16. y′=[1t−1−e−tt−1ett+11t+1]y+[t2−1t2−1];Y=t[te−tett]
17. y′=1t[110021−222]y+[121];Y=[t2t31t22t3−102t32]
18. y′=[3ete2te−t2ete−2te−t1]y+[e3t00];Y=[e5te2t0e4t0ete3t−1−1]
19. y′=1t[1t001t0−t1]y+[ttt];Y=t[1costsint0−sintcost0−cost−sint]
20. y′=−1t[e−t−t1−e−te−t1−t−e−te−t−t1−e−t]y+1t[et0et];Y=1t[ete−ttet−e−te−tete−t0]
21. Prove Theorem 10.7.1.
22.
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)(Y−1(t0)k+∫tt0Y−1(s)f(s)ds).