Section 7.3

  1. 1. x1(t)=c1e−t+c2e3t, x2(t)=−c1e−t+c2e3tx1(t)=c1e−t+c2e3t, x2(t)=−c1e−t+c2e3t

  2. 2. x1(t)=c1e−t+3c2e4t, x2(t)=−c1e−t+2c2e4tx1(t)=c1e−t+3c2e4t, x2(t)=−c1e−t+2c2e4t

  3. 3. General solution x1(t)=c1e−t+4c2e6t, x2(t)=−c1e−t+3c2e6tx1(t)=c1e−t+4c2e6t, x2(t)=−c1e−t+3c2e6t Particular solution x1(t)=17(−e−t+8e6t), x2(t)=17(e−t+6e6t)x1(t)=17(−e−t+8e6t), x2(t)=17(e−t+6e6t).

  4. 4. x1(t)=c1e−2t+c2e5t, x2(t)=−6c1e−2t+c2e5tx1(t)=c1e−2t+c2e5t, x2(t)=−6c1e−2t+c2e5t

  5. 5. x1(t)=c1e−t+7c2e5t, x2(t)=c1e−t+c2e5tx1(t)=c1e−t+7c2e5t, x2(t)=c1e−t+c2e5t

  6. 6. General solution x1(t)=5c1e3t+c2e4tx1(t)=5c1e3t+c2e4t, x2(t)=−6c1e3t−c2e4tx2(t)=−6c1e3t−c2e4t Particular solution x1(t)=−5e3t+6e4tx1(t)=−5e3t+6e4t, x2(t)=6e3t−6e4tx2(t)=6e3t−6e4t.

  7. 7. x1(t)=c1et+2c2e−9t, x2(t)=c1et−3c2e−9tx1(t)=c1et+2c2e−9t, x2(t)=c1et−3c2e−9t

  8. 8. x1(t)=5c1 cos 2t+5c2 sin 2tx1(t)=5c1 cos 2t+5c2 sin 2t, x2(t)=(c1−2c2) cos 2t+(2c1+c2) sin 2tx2(t)=(c1−2c2) cos 2t+(2c1+c2) sin 2t

  9. 9. General solution x1(t)=5c1 cos 4t+5c2 sin 4tx1(t)=5c1 cos 4t+5c2 sin 4t, x2(t)=c1(2 cos 4t+4 sin 4t)+c2(2 sin 4t−4 cos 4t)x2(t)=c1(2 cos 4t+4 sin 4t)+c2(2 sin 4t−4 cos 4t). Particular solution x1(t)=2 cos 4t−114sin 4tx1(t)=2 cos 4t−114sin 4t, x2(t)=3 cos 4t+12sin 4tx2(t)=3 cos 4t+12sin 4t

  10. 10. x1(t)=−2c1 cos 3t−2c2 sin 3tx1(t)=−2c1 cos 3t−2c2 sin 3t, x2(t)=(3c1+3c2) cos 3t+(3c2−3c1) sin 3tx2(t)=(3c1+3c2) cos 3t+(3c2−3c1) sin 3t

  11. 11. General solution x1(t)=et(c1 cos 2t−c2 sin 2t)x1(t)=et(c1 cos 2t−c2 sin 2t), x2(t)=et(c1 sin 2t+c2 cos 2t)x2(t)=et(c1 sin 2t+c2 cos 2t) Particular solution x1(t)=−4et sin 2tx1(t)=−4et sin 2t, x2(t)=4et cos 2tx2(t)=4et cos 2t

  12. 12. x1(t)=e2t(−5c1 cos 2t−5c2 sin 2t)x1(t)=e2t(−5c1 cos 2t−5c2 sin 2t), x2(t)=e2t[(c1+2c2) cos 2t+(−2c1+c2) sin 2t]x2(t)=e2t[(c1+2c2) cos 2t+(−2c1+c2) sin 2t]

  13. 13. x1(t)=3e2t(c1 cos 3t−c2 sin 3t)x1(t)=3e2t(c1 cos 3t−c2 sin 3t), x2(t)=e2t[(c1+c2) cos 3t+(c1−c2) sin 3t]x2(t)=e2t[(c1+c2) cos 3t+(c1−c2) sin 3t]

  14. 14. x1(t)=e3t(c1 cos 4t+c2 sin 4t), x2(t)=e3t(c1 sin 4t−c2 cos 4t)x1(t)=e3t(c1 cos 4t+c2 sin 4t), x2(t)=e3t(c1 sin 4t−c2 cos 4t)

  15. 15. x1(t)=5e5t(c1 cos 4t−c2 sin 4t), x2(t)=e5t[(2c1+4c2) cos 4t+(4c1−2c2) sin 4t]x1(t)=5e5t(c1 cos 4t−c2 sin 4t), x2(t)=e5t[(2c1+4c2) cos 4t+(4c1−2c2) sin 4t]

  16. 16. x1(t)=c1e−10t+2c2e−100t, x2(t)=2c1e−10t−5c2e−100tx1(t)=c1e−10t+2c2e−100t, x2(t)=2c1e−10t−5c2e−100t

  17. 17. x1(t)=c1e9t+c2e6t+c3, x2(t)=c1e9t−2c2e6t, x3(t)=c1e9t+c2e6t−c3x1(t)=c1e9t+c2e6t+c3, x2(t)=c1e9t−2c2e6t, x3(t)=c1e9t+c2e6t−c3

  18. 18. x1(t)=c1e9t+4c3, x2(t)=2c1e9t+c2e6t−c3, x3(t)=2c1e9t−c2e6t−c3x1(t)=c1e9t+4c3, x2(t)=2c1e9t+c2e6t−c3, x3(t)=2c1e9t−c2e6t−c3

  19. 19. x1(t)=c1e6t+c2e3t+c3e3tx1(t)=c1e6t+c2e3t+c3e3t, x2(t)=c1e6t−2c2e3t, x3(t)=c1e6t+c2e3t−c3e3tx2(t)=c1e6t−2c2e3t, x3(t)=c1e6t+c2e3t−c3e3t

  20. 20. x1(t)=c1e9t+c2e6t+c3e2t, x2(t)=c1e9t−2c2e6t, x3(t)=c1e9t+c2e6t−c3e2tx1(t)=c1e9t+c2e6t+c3e2t, x2(t)=c1e9t−2c2e6t, x3(t)=c1e9t+c2e6t−c3e2t

  21. 21. x1(t)=6c1+3c2et+2c3e−t, x2(t)=2c1+c2et+c3e−t, x3(t)=5c1+2c2et+2c3e−tx1(t)=6c1+3c2et+2c3e−t, x2(t)=2c1+c2et+c3e−t, x3(t)=5c1+2c2et+2c3e−t

  22. 22. x1(t)=c2et+c3e3t, x2(t)=c1e−2t−c2et−c3e3t, x3(t)=−c1e−2t+c3e3tx1(t)=c2et+c3e3t, x2(t)=c1e−2t−c2et−c3e3t, x3(t)=−c1e−2t+c3e3t

  23. 23. x1(t)=c1e2t+c3e3tx1(t)=c1e2t+c3e3t, x2(t)=−c1e2t+c2e−2t−c3e3t, x3(t)=−c2e−2t+c3e3tx2(t)=−c1e2t+c2e−2t−c3e3t, x3(t)=−c2e−2t+c3e3t

  24. 24. x1(t)=c1et+c2(2 cos 2t− sin 2t)+c3(cos 2t+2 sin 2t) x2(t)=−c1et−c2(3 cos 2t+ sin 2t)+c3(cos 2t−3 sin 2t) x3(t)=c2(3 cos 2t+ sin 2t)+c3(3 sin 2t− cos 2t)x1(t)=c1et+c2(2 cos 2t− sin 2t)+c3(cos 2t+2 sin 2t) x2(t)=−c1et−c2(3 cos 2t+ sin 2t)+c3(cos 2t−3 sin 2t) x3(t)=c2(3 cos 2t+ sin 2t)+c3(3 sin 2t− cos 2t)

  25. 25. x1(t)=c1+e2t[(c2+c3) cos 3t+(−c2+c3) sin 3t], x2(t)=−c1+2e2t(−c2 cos 3t−c3 sin 3t), x3(t)=2e2t(c2 cos 3t+c3 sin 3t)x1(t)=c1+e2t[(c2+c3) cos 3t+(−c2+c3) sin 3t], x2(t)=−c1+2e2t(−c2 cos 3t−c3 sin 3t), x3(t)=2e2t(c2 cos 3t+c3 sin 3t)

  26. 26. x1(t)=4e3t−e−t(4 cos t−sin t),  x2(t)=9e3t−e−t(9 cos t+2 sin t), x3(t)=17e−t cos tx1(t)=4e3t−e−t(4 cos t−sin t),  x2(t)=9e3t−e−t(9 cos t+2 sin t), x3(t)=17e−t cos t

  27. 27. x1(t)=15e−0.2t, x2(t)=15(e−0.2t−e−0.4t).x1(t)=15e−0.2t, x2(t)=15(e−0.2t−e−0.4t). The maximum amount ever in tank 2 is x2(5 ln 2)=3.75 lb.x2(5 ln 2)=3.75 lb.

  28. 28. x1(t)=15e−0.4t, x2(t)=40(−e−0.4t+e−0.25t)x1(t)=15e−0.4t, x2(t)=40(−e−0.4t+e−0.25t). The maximum amount ever in tank 2 is about 6.85 lb.

  29. 29. x1(t)=10+5e−0.6t, x2(t)=5−5e−0.6tx1(t)=10+5e−0.6t, x2(t)=5−5e−0.6t

  30. 30. x1(t)=15c1+c2e−0.65t,x2(t)=8c1−c2e−0.65t.x1(t)=15c1+c2e−0.65t,x2(t)=8c1−c2e−0.65t.

  31. 31. x1(t)=27e−t, x2(t)=27e−t−27e−2t, x3(t)=27e−t−54e−2t+27e−3tx1(t)=27e−t, x2(t)=27e−t−27e−2t, x3(t)=27e−t−54e−2t+27e−3t.

    The maximum amount of salt everin tank 3 is x3(ln 3)=4 poundsx3(ln 3)=4 pounds.

  32. 32. x1(t)=45e−3tx1(t)=45e−3t, x2(t)=−135e−3t+135e−2tx2(t)=−135e−3t+135e−2t, x3(t)=135e−3t−270e−2t+135e−tx3(t)=135e−3t−270e−2t+135e−t.

    The maximum amount of salt ever in tank 3 is x3(ln 3)=20 poundsx3(ln 3)=20 pounds.

  33. 33. x1(t)=45e−4tx1(t)=45e−4t, x2(t)=90e−4t−90e−6t, x3(t)=−270e−4t+135e−6t+135e−2tx2(t)=90e−4t−90e−6t, x3(t)=−270e−4t+135e−6t+135e−2t.

    The maximum amount of saltever in tank 3 is x3(12 ln 3)=20x3(12 ln 3)=20 pounds.

  34. 34. x1(t)=40e−3t, x2(t)=60e−3t−60e−5t, x3(t)=−150e−3t+75e−5t+75e−tx1(t)=40e−3t, x2(t)=60e−3t−60e−5t, x3(t)=−150e−3t+75e−5t+75e−t.

    The maximum amount of salt ever in tank 3 is x3(12 ln 5)≈21.4663x3(12 ln 5)≈21.4663 pounds.

  35. 35. x1(t)=10−17(55e−18t−216e−11t), x2(t)=3−17(165e−18t−144e−11t), x3(t)=20+17(220e−18t−360e−11t)x1(t)=10−17(55e−18t−216e−11t), x2(t)=3−17(165e−18t−144e−11t), x3(t)=20+17(220e−18t−360e−11t). The limiting amounts of salt in tanks 1, 2, and 3 are 10 lb, 3 lb, and 20 lb.

  36. 36. x1(t)=4+e−3t/5[14 cos (3t/10)−2 sin (3t/10)], x2(t)=10−e−3t/5[10 cos (3t/10)−10 sin (3t/10)], x3(t)=4−e−3t/5[4 cos (3t/10)+8 sin (3t/10)]x1(t)=4+e−3t/5[14 cos (3t/10)−2 sin (3t/10)], x2(t)=10−e−3t/5[10 cos (3t/10)−10 sin (3t/10)], x3(t)=4−e−3t/5[4 cos (3t/10)+8 sin (3t/10)]. The limiting amounts of salt in tanks 1, 2, and 3 are 4 lb, 10 lb, and 4 lb.

  37. 37. x1(t)=30+e−3t[25 cos (t√2)+10√2 sin (t√2)], x2(t)=10−e−3t[10 cos (t√2)−252√2 sin (t√2)], x3(t)=15−e−3t[15 cos (t√2)+452√2 sin (t√2)]x1(t)=30+e−3t[25 cos (t2–√)+102–√ sin (t2–√)], x2(t)=10−e−3t[10 cos (t2–√)−2522–√ sin (t2–√)], x3(t)=15−e−3t[15 cos (t2–√)+4522–√ sin (t2–√)]. The limiting amounts of salt in tanks 1, 2, and 3 are 30 lb, 10 lb, and 15 lb.

  38. 38. x1(t)=c1etx1(t)=c1et, x2(t)=−2c1et+c2e2tx2(t)=−2c1et+c2e2t, x3(t)=3c1et−3c2e2t+c3e3tx3(t)=3c1et−3c2e2t+c3e3t, x4(t)=−4c1et+6c2e2t−4c3e3t+c4e4tx4(t)=−4c1et+6c2e2t−4c3e3t+c4e4t

  39. 39. x1(t)=3c1et+c4e−2t, x2(t)=−2c1et+c3e2t−c4e−2tx1(t)=3c1et+c4e−2t, x2(t)=−2c1et+c3e2t−c4e−2t, x3(t)=4c1et+c2e−t, x4(t)=c1etx3(t)=4c1et+c2e−t, x4(t)=c1et

  40. 40. x1(t)=c1e2tx1(t)=c1e2t, x2(t)=−3c1e2t+3c2e−2t−c4e−5tx2(t)=−3c1e2t+3c2e−2t−c4e−5t, x3(t)=c3e5t, x4(t)=−c2e−2t−3c3e5tx3(t)=c3e5t, x4(t)=−c2e−2t−3c3e5t

  41. 41. x1(t)=2e10t+e15t=x4(t), x2(t)=−e10t+2e15t=x3(t)x1(t)=2e10t+e15t=x4(t), x2(t)=−e10t+2e15t=x3(t)

  42. 42. x(t)=c1[3−12]+c2[111]e2t+c3[2−31]e5tx(t)=c1⎡⎣⎢3−12⎤⎦⎥+c2⎡⎣⎢111⎤⎦⎥e2t+c3⎡⎣⎢2−31⎤⎦⎥e5t

  43. 43. x(t)=c1[3−15]e−2t+c2[111]e4t+c3[1−13]e8tx(t)=c1⎡⎣⎢3−15⎤⎦⎥e−2t+c2⎡⎣⎢111⎤⎦⎥e4t+c3⎡⎣⎢1−13⎤⎦⎥e8t

  44. 44. x(t)=c1[3−22]e−3t+c2[715]e6t+c3[5−33]e12tx(t)=c1⎡⎣⎢3−22⎤⎦⎥e−3t+c2⎡⎣⎢715⎤⎦⎥e6t+c3⎡⎣⎢5−33⎤⎦⎥e12t

  45. 45. x(t)=c1[111−1]e−3t+c2[12−11]+c3[2111]e3t+c4[1−12−1]e6tx(t)=c1⎡⎣⎢⎢⎢⎢111−1⎤⎦⎥⎥⎥⎥e−3t+c2⎡⎣⎢⎢⎢⎢12−11⎤⎦⎥⎥⎥⎥+c3⎡⎣⎢⎢⎢⎢2111⎤⎦⎥⎥⎥⎥e3t+c4⎡⎣⎢⎢⎢⎢1−12−1⎤⎦⎥⎥⎥⎥e6t

  46. 46. x(t)=c1[32−11]e−4t+c2[122−1]e2t+c3[11−11]e4t+c4[3−23−3]e8tx(t)=c1⎡⎣⎢⎢⎢⎢32−11⎤⎦⎥⎥⎥⎥e−4t+c2⎡⎣⎢⎢⎢⎢122−1⎤⎦⎥⎥⎥⎥e2t+c3⎡⎣⎢⎢⎢⎢11−11⎤⎦⎥⎥⎥⎥e4t+c4⎡⎣⎢⎢⎢⎢3−23−3⎤⎦⎥⎥⎥⎥e8t

  47. 47. x(t)=c1[221−1]e−3t+c2[12−11]e3t+c3[2111]e6t+c4[1−12−1]e9tx(t)=c1⎡⎣⎢⎢⎢⎢221−1⎤⎦⎥⎥⎥⎥e−3t+c2⎡⎣⎢⎢⎢⎢12−11⎤⎦⎥⎥⎥⎥e3t+c3⎡⎣⎢⎢⎢⎢2111⎤⎦⎥⎥⎥⎥e6t+c4⎡⎣⎢⎢⎢⎢1−12−1⎤⎦⎥⎥⎥⎥e9t

  48. 48. x(t)=c1[12−12]e16t+c2[251−1]e32t+c3[3−112]e48t+c4[112−3]e64tx(t)=c1⎡⎣⎢⎢⎢⎢12−12⎤⎦⎥⎥⎥⎥e16t+c2⎡⎣⎢⎢⎢⎢251−1⎤⎦⎥⎥⎥⎥e32t+c3⎡⎣⎢⎢⎢⎢3−112⎤⎦⎥⎥⎥⎥e48t+c4⎡⎣⎢⎢⎢⎢112−3⎤⎦⎥⎥⎥⎥e64t

  49. 49. x(t)=c1[10311]e−3t+c2[030−11]+c3[17111]e3t+c4[01011]e6t+c5[20521]e9tx(t)=c1⎡⎣⎢⎢⎢⎢⎢⎢10311⎤⎦⎥⎥⎥⎥⎥⎥e−3t+c2⎡⎣⎢⎢⎢⎢⎢⎢030−11⎤⎦⎥⎥⎥⎥⎥⎥+c3⎡⎣⎢⎢⎢⎢⎢⎢17111⎤⎦⎥⎥⎥⎥⎥⎥e3t+c4⎡⎣⎢⎢⎢⎢⎢⎢01011⎤⎦⎥⎥⎥⎥⎥⎥e6t+c5⎡⎣⎢⎢⎢⎢⎢⎢20521⎤⎦⎥⎥⎥⎥⎥⎥e9t

  50. 50. x(t)=c1[011101]e−7t+c2[100011]e−4t+c3[010101]e3t+c4[001010]e5t+c5[110001]e9t+[001−1−10]e11tx(t)=c1⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢011101⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥e−7t+c2⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢100011⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥e−4t+c3⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢010101⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥e3t+c4⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢001010⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥e5t+c5⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢110001⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥e9t+⎡⎣⎢⎢⎢⎢⎢⎢⎢⎢⎢001−1−10⎤⎦⎥⎥⎥⎥⎥⎥⎥⎥⎥e11t

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