Section 8.3
1. x(t)=12[e−t+e3t−e−t+e3t−e−t+e3te−t+e3t]
x(t)=12[e−t+e3t−e−t+e3t−e−t+e3te−t+e3t] 2. x=15[2e−t+3e4t−3e−t+3e4t−2e−t+2e4t3e−t+2e4t]
x=15[2e−t+3e4t−2e−t+2e4t−3e−t+3e4t3e−t+2e4t] 3. x(t)=17[3e−t+4e6t−4e−t+4e6t−3e−t+3e6t4e−t+3e6t]
x(t)=17[3e−t+4e6t−3e−t+3e6t−4e−t+4e6t4e−t+3e6t] 4. x(t)=17[e−2t+6e5t−e−2t+e5t−6e−2t+6e5t6e−2t+e5t]
x(t)=17[e−2t+6e5t−6e−2t+6e5t−e−2t+e5t6e−2t+e5t] 5. x(t)=16[−e−t+7e5t7e−t−7e5t−e−t+e5t7e−t−e5t]
x(t)=16[−e−t+7e5t−e−t+e5t7e−t−7e5t7e−t−e5t] 6. x(t)=[−5e3t+6e4t−5e3t+5e4t6e3t−6e4t6e3t−5e4t]
x(t)=[−5e3t+6e4t6e3t−6e4t−5e3t+5e4t6e3t−5e4t] 7. x(t)=15[2e−9t+3et−2e−9t+2et−3e−9t+3et3e−9t+2et]
x(t)=15[2e−9t+3et−3e−9t+3et−2e−9t+2et3e−9t+2et] 8. x(t)=12[2 cos 2t+sin 2t−5 sin 2tsin 2t2 cos 2t−sin 2t]
x(t)=12[2 cos 2t+sin 2tsin 2t−5 sin 2t2 cos 2t−sin 2t] 9. x(t)=14[4 cos 4t+2 sin 4t−5 sin 4t4 sin 4t4 cos 4t−2 sin 4t]
x(t)=14[4 cos 4t+2 sin 4t4 sin 4t−5 sin 4t4 cos 4t−2 sin 4t] 10. x(t)=13[3 cos 3t−3 sin 3t−2 sin 3t9 sin 3t3 cos 3t+3 sin 3t]
x(t)=13[3 cos 3t−3 sin 3t9 sin 3t−2 sin 3t3 cos 3t+3 sin 3t] 11. x(t)=et[cos 2t−sin 2tsin 2tcos 2t]
x(t)=et[cos 2tsin 2t−sin 2tcos 2t] 12. x(t)=e2t2[2 cos 2t−sin 2t−5 sin 2tsin 2t2 cos 2t+sin 2t]
x(t)=e2t2[2 cos 2t−sin 2tsin 2t−5 sin 2t2 cos 2t+sin 2t] 13. x(t)=e2t3[3 cos 3t+3 sin 3t−9 sin 3t2 sin 3t3 cos 3t−3 sin 3t]
x(t)=e2t3[3 cos 3t+3 sin 3t2 sin 3t−9 sin 3t3 cos 3t−3 sin 3t] 14. x(t)=e3t[cos 4t−sin 4tsin 4tcos 4t]
x(t)=e3t[cos 4tsin 4t−sin 4tcos 4t] 15. x(t)=e5t4[4 cos 4t+2 sin 4t−5 sin 4t4 sin 4t4 cos 4t−2 sin 2t]
x(t)=e5t4[4 cos 4t+2 sin 4t4 sin 4t−5 sin 4t4 cos 4t−2 sin 2t] 16. x(t)=19[4e−100t+5e−10t−2e−100t+2e−10t−10e−100t+10e−10t5e−100t+4e−10t]
x(t)=19[4e−100t+5e−10t−10e−100t+10e−10t−2e−100t+2e−10t5e−100t+4e−10t] 17. x(t)=16[3+e6t+2e9t−2e6t+2e9t−3+e6t+2e9t−2e6t+2e9t4e6t+2e9t−2e6t+2e9t−3+e6t+2e9t−2e6t+2e9t3+e6t+2e9t]
x(t)=16⎡⎣⎢3+e6t+2e9t−2e6t+2e9t−3+e6t+2e9t−2e6t+2e9t4e6t+2e9t−2e6t+2e9t−3+e6t+2e9t−2e6t+2e9t3+e6t+2e9t⎤⎦⎥ 18. x(t)=118[16+e6t+2e9t−4+4e9t−4+4e9t−4e6t+4e9t1+9e6t+8e9t1−9e6t+8e9t−4e6t+4e9t1−9e6t+8e9t1+9e6t+8e9t]
x(t)=118⎡⎣⎢16+e6t+2e9t−4e6t+4e9t−4e6t+4e9t−4+4e9t1+9e6t+8e9t1−9e6t+8e9t−4+4e9t1−9e6t+8e9t1+9e6t+8e9t⎤⎦⎥ 19. x(t)=13[2e3t+e6t−e3t+e6t−e3t+e6t−e3t+e6t2e3t+e6t−e3t+e6t−e3t+e6t−e3t+e6t2e3t+e6t]
x(t)=13⎡⎣⎢2e3t+e6t−e3t+e6t−e3t+e6t−e3t+e6t2e3t+e6t−e3t+e6t−e3t+e6t−e3t+e6t2e3t+e6t⎤⎦⎥ 20. x(t)=16[3e2t+e6t+2e9t−2e6t+2e9t−3e2t+e6t+2e9t−2e6t+e9t4e6t+2e9t−2e6t+e9t−3e2t+e6t+2e9t−2e6t+e9t3e2t+e6t+2e9t]
x(t)=16⎡⎣⎢3e2t+e6t+2e9t−2e6t+e9t−3e2t+e6t+2e9t−2e6t+2e9t4e6t+2e9t−2e6t+e9t−3e2t+e6t+2e9t−2e6t+e9t3e2t+e6t+2e9t⎤⎦⎥
21. x(t)=e−3t[1+tt−t1−t]
x(t)=e−3t[1+t−tt1−t] 22. x(t)=e2t[1+t−tt1−t]
x(t)=e2t[1+tt−t1−t] 23. x(t)=e3t[1−2t−2t2t1+2t]
x(t)=e3t[1−2t2t−2t1+2t] 24. x(t)=e4t[1−t−tt1+t]
x(t)=e4t[1−tt−t1+t] 25. x(t)=e5t[1+2tt−4t1−2t]
x(t)=e5t[1+2t−4tt1−2t] 26. x(t)=e5t[1−4t−4t4t1+4t]
x(t)=e5t[1−4t4t−4t1+4t] 27. x(t)=[e2t00e2t−e9te9t−e2t+e9t00e2t]
x(t)=⎡⎣⎢e2te2t−e9t00e9t00−e2t+e9te2t⎤⎦⎥ 28. x(t)=[−2e7t+3e13t−2e7t+2e13t03e7t−3e13t3e7t−2e13t0−e7t+e13t−e7t+e13te13t]
x(t)=⎡⎣⎢−2e7t+3e13t3e7t−3e13t−e7t+e13t−2e7t+2e13t3e7t−2e13t−e7t+e13t00e13t⎤⎦⎥ 29. x(t)=[7e5t−6e9t−3e5t+3e9t−21e5t+21e9t0e5t02e5t−2e9t−e5t+e9t−6e5t+7e9t]
x(t)=⎡⎣⎢7e5t−6e9t02e5t−2e9t−3e5t+3e9te5t−e5t+e9t−21e5t+21e9t0−6e5t+7e9t⎤⎦⎥ 30. x(t)=[5e3t−4e7t−10e3t+10e7t12e3t−12e7t2e3t−2e7t−4e3t+5e7t6e3t−6e7t00e3t]
x(t)=⎡⎣⎢5e3t−4e7t2e3t−2e7t0−10e3t+10e7t−4e3t+5e7t012e3t−12e7t6e3t−6e7te3t⎤⎦⎥ 31. eAt=14[−e−t+5e3te−t−e3t−5e−t+5e3t5e−t−e3t], x(t)=[−e−t−14e2t+15e3t−5e−t−10e2t+15e3t]
eAt=14[−e−t+5e3t−5e−t+5e3te−t−e3t5e−t−e3t], x(t)=[−e−t−14e2t+15e3t−5e−t−10e2t+15e3t] 32. With eAt
eAt as in Problem 31 , x(t)=[(−10−7t)e−t+(10−5t)e3t(−15−35t)e−t+(15−5t)e3t].x(t)=[(−10−7t)e−t+(10−5t)e3t(−15−35t)e−t+(15−5t)e3t]. 33. eAt=[1+3t−t9t1−3t], x(t)=[3+11t+8t25+17t+24t2]
eAt=[1+3t9t−t1−3t], x(t)=[3+11t+8t25+17t+24t2] 34. With eAt
eAt as in Problem 33 , x(t)=[2+t+ln t5+3t−1t+3 ln t].x(t)=[2+t+ln t5+3t−1t+3 ln t]. 35. eAt=[cos t+ 2 sin t−5 sin tsin tcos t− 2 sin t], x(t)=[−1+8t+cos t−8 sin t−2+4t+2 cos t−3 sin t]
eAt=[cos t+ 2 sin tsin t−5 sin tcos t− 2 sin t], x(t)=[−1+8t+cos t−8 sin t−2+4t+2 cos t−3 sin t] 36. With eAt
eAt as in Problem 35 , x(t)=[3 cos t−32 sin t+17t cos t+4t sin t5 cos t−13 sin t+6t cos t+5t sin t].x(t)=[3 cos t−32 sin t+17t cos t+4t sin t5 cos t−13 sin t+6t cos t+5t sin t]. 37. eAt=[1+2t−4tt1−2t], x(t)=[8t3+6t43t2−2t3+3t4]
eAt=[1+2tt−4t1−2t], x(t)=[8t3+6t43t2−2t3+3t4] 38. With eAt
eAt as in Problem 37 , x(t)=[−7+14t−6t2+4t2 ln t−7+9t−3t2+ln t−2t ln t+2t2 ln t]x(t)=[−7+14t−6t2+4t2 ln t−7+9t−3t2+ln t−2t ln t+2t2 ln t] .39. eAt=[cos t−sin tsin tcos t], x(t)=[t cos t−(ln cos t)(sin t)t sin t+(ln cos t)(cos t)]
eAt=[cos tsin t−sin tcos t], x(t)=[t cos t−(ln cos t)(sin t)t sin t+(ln cos t)(cos t)] 40. eAt=[cos 2t−2 sin 2tsin 2tcos 2t], x(t)=[12t2 cos 2t12t2 sin 2t]
eAt=[cos 2tsin 2t−2 sin 2tcos 2t], x(t)=[12t2 cos 2t12t2 sin 2t] 41. x(t)=[−9e−t+10e3t−2e−t+2e3t4e−t−4e3t9e−t−9e3t2e−t−2e3t−4e−t+4e3t−18e−t+18e3t−4e−t+4e3t8e−t−7e3t]
x(t)=⎡⎣⎢−9e−t+10e3t9e−t−9e3t−18e−t+18e3t−2e−t+2e3t2e−t−2e3t−4e−t+4e3t4e−t−4e3t−4e−t+4e3t8e−t−7e3t⎤⎦⎥ 42. x(t)=[−5e−2t+6e3t−10e−2t+10e3t−20e−2t+20e3t−3e−2t+3e3t−6e−2t+7e3t−12e−2t+12e3t3e−2t+−3e3t6e−2t−6e3t12e−2t−11e3t]
x(t)=⎡⎣⎢−5e−2t+6e3t−3e−2t+3e3t3e−2t+−3e3t−10e−2t+10e3t−6e−2t+7e3t6e−2t−6e3t−20e−2t+20e3t−12e−2t+12e3t12e−2t−11e3t⎤⎦⎥ 43. x(t)=12e2t[−t2−8t+24t2+34tt2+8t−2t8t+22t−t24t2+2tt2+2]
x(t)=12e2t⎡⎣⎢−t2−8t+2−2t−t24t2+34t8t+24t2+2tt2+8t2tt2+2⎤⎦⎥ 44. x(t)=12e3t[4t+2−2t2t2t2+2t−t2+2t22t2−6t−t2+4tt2−4t+2]
x(t)=12e3t⎡⎣⎢4t+22t2+2t2t2−6t−2t−t2+2−t2+4t2tt2t2−4t+2⎤⎦⎥ 45. x(t)=[e−tte−tte−t−2te−t−3e−t+(3−2t)e2t(1−3t)e−t(1−3t)e−t−e2t2(3t−1)e−t+(2−t)e2t−e−t+(1+2t)e2t−te−t−te−t+e2t2te−t+te2t−2e−t+2e2t−2te−t−2te−t4te−t+e2t]
x(t)=⎡⎣⎢⎢⎢⎢e−t−3e−t+(3−2t)e2t−e−t+(1+2t)e2t−2e−t+2e2tte−t(1−3t)e−t−te−t−2te−tte−t(1−3t)e−t−e2t−te−t+e2t−2te−t−2te−t2(3t−1)e−t+(2−t)e2t2te−t+te2t4te−t+e2t⎤⎦⎥⎥⎥⎥ 46. x(t)=12et[48t2+68t+2−18t2−24t6t2+8t36t2+60t7t2+44t−3t2−18t+2t2+6t6t2+38t−21t2−20t9t2+6t−3t2−2t+2−18t2−18t−42t2−54t18t2+18t−6t2−6t−36t2−48t+2]
x(t)=12et⎡⎣⎢⎢⎢⎢48t2+68t+27t2+44t−21t2−20t−42t2−54t−18t2−24t−3t2−18t+29t2+6t18t2+18t6t2+8tt2+6t−3t2−2t+2−6t2−6t36t2+60t6t2+38t−18t2−18t−36t2−48t+2⎤⎦⎥⎥⎥⎥ 47. x(t)=[cos tcos t]+i[−sin 3tsin 3t]
x(t)=[cos tcos t]+i[−sin 3tsin 3t] There are two natural modes—one in which the two masses move in the same direction with frequency ω1=1ω1=1 and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=3ω2=3 and with equal amplitudes.48. x(t)=[cos tcos t]+i[−sin √5 tsin √5 t]
x(t)=[cos tcos t]+i[−sin 5–√ tsin 5–√ t] There are two natural modes—one in which the two masses move in the same direction with frequency ω1=1ω1=1 and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=√5ω2=5–√ and with equal amplitudes.49. x(t)=[cos √2 tcos √2 t]+i[−sin 2tsin 2t]
x(t)=[cos 2–√ tcos 2–√ t]+i[−sin 2tsin 2t] There are two natural modes—one in which the two masses move in the same direction with frequency ω1=√2ω1=2–√ and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=2ω2=2 and with equal amplitudes.50. x(t)=[cos 2tcos 2t]+i[−sin 4tsin 4t]
x(t)=[cos 2tcos 2t]+i[−sin 4tsin 4t] There are two natural modes-one in which the two masses move in the same direction with frequency ω1=2ω1=2 and with equal amplitudes, and one in which they move in opposite directions with frequency ω2=4ω2=4 and with equal amplitudes.