Uppgift 5.3: 11 

restart 

TvÃ¥ ekvationer, speciellt av grad>1, hanterar man nog lika enkelt med dsolve som med matriser. 

`:=`(ode, ((`@@`(D, 2))(x))(t) = `+`(`-`(`*`(40, `*`(x(t)))), `*`(8, `*`(y(t))), f[1]), ((`@@`(D, 2))(y))(t) = `+`(`*`(12, `*`(x(t))), `-`(`*`(60, `*`(y(t)))), f[2])) 

((`@@`(D, 2))(x))(t) = `+`(`-`(`*`(40, `*`(x(t)))), `*`(8, `*`(y(t))), f[1]), ((`@@`(D, 2))(y))(t) = `+`(`*`(12, `*`(x(t))), `-`(`*`(60, `*`(y(t)))), f[2]) (5.1)
 

`:=`(ic, x(0) = 19, (D(x))(0) = 12, y(0) = 3, (D(y))(0) = 6) 

x(0) = 19, (D(x))(0) = 12, y(0) = 3, (D(y))(0) = 6 (5.2)
 

`:=`(ode_a, eval(ode, {f[1] = 0, f[2] = 0})) 

((`@@`(D, 2))(x))(t) = `+`(`-`(`*`(40, `*`(x(t)))), `*`(8, `*`(y(t)))), ((`@@`(D, 2))(y))(t) = `+`(`*`(12, `*`(x(t))), `-`(`*`(60, `*`(y(t))))) (5.3)
 

`:=`(sol_a, dsolve({ic, ode_a})) 

{x(t) = `+`(`*`(2, `*`(sin(`+`(`*`(6, `*`(t)))))), `*`(`/`(120, 7), `*`(cos(`+`(`*`(6, `*`(t)))))), `*`(`/`(13, 7), `*`(cos(`+`(`*`(8, `*`(t))))))), y(t) = `+`(sin(`+`(`*`(6, `*`(t)))), `*`(`/`(60, 7)... (5.4)
 

Klipp och klistra ur sol_a lösningskomponenterna med frekvenserna 6 resp, 8 och rita dem med plot. 

 

Plot_2d
 

 

Plot_2d
 

`:=`(ode_b, eval(ode, {f[1] = `+`(`-`(`*`(195, `*`(cos(`+`(`*`(7, `*`(t)))))))), f[2] = `+`(`-`(`*`(195, `*`(cos(`+`(`*`(7, `*`(t))))))))})) 

((`@@`(D, 2))(x))(t) = `+`(`-`(`*`(40, `*`(x(t)))), `*`(8, `*`(y(t))), `-`(`*`(195, `*`(cos(`+`(`*`(7, `*`(t)))))))), ((`@@`(D, 2))(y))(t) = `+`(`*`(12, `*`(x(t))), `-`(`*`(60, `*`(y(t)))), `-`(`*`(19... (5.5)
 

`:=`(sol_b, dsolve({ic, ode_b})) 

{x(t) = `+`(`*`(19, `*`(cos(`+`(`*`(7, `*`(t)))))), `*`(2, `*`(sin(`+`(`*`(6, `*`(t))))))), y(t) = `+`(`*`(3, `*`(cos(`+`(`*`(7, `*`(t)))))), sin(`+`(`*`(6, `*`(t)))))} (5.6)
 

Klipp ur och klistra in komponenter med samma frekvens i plot. 

 

Plot_2d
 

 

Plot_2d