# No Interpretations Available When Solving an inhomogeneous Second Linear ODE System(a Forced Vibration Problem)

My input is: DSolve[{m*n1''[t] + k*n1[t] - k*n2[t] == f*E^(I*w*t), m*n2''[t] - k*n1[t] + 2*k*n2[t] - k*n3[t] == 0, m*n3''[t] - k*n2[t] + k*n3[t] == 0}, {n1[t], n2[t], n3[t]}, t] What's wrong with me? Thanks.

• E^[I*w*t] should read: E^(I*w*t) Commented Nov 30, 2022 at 19:13
• I changed into E^(Iwt) but still failed. Commented Nov 30, 2022 at 19:17

\$Version

(* "13.1.0 for Mac OS X x86 (64-bit) (June 16, 2022)" *)

Clear["Global*"]

eqns = {m*n1''[t] + k*n1[t] - k*n2[t] == f*E^(I*w*t),
m*n2''[t] - k*n1[t] + 2*k*n2[t] - k*n3[t] == 0,
m*n3''[t] - k*n2[t] + k*n3[t] == 0};

(sol = DSolve[eqns, {n1, n2, n3}, t]) // Short[#, 5] &


Verifying the solutions,

eqns /. sol // Simplify

(* {{True, True, True}} *)


Simplifying the solutions,

solt = {n1[t], n2[t], n3[t]} /. sol // FullSimplify

(* {{1/18 (6 (-((3 E^(I t w) f (k^2 - 3 k m w^2 + m^2 w^4))/(
m w^2 (3 k^2 - 4 k m w^2 + m^2 w^4))) + C[1] + C[3] + C[5] +
t (C[2] + C[4] + C[6])) + 9 (C[1] - C[5]) Cos[(Sqrt[k] t)/Sqrt[m]] +
3 (C[1] - 2 C[3] + C[5]) Cos[(Sqrt[3] Sqrt[k] t)/Sqrt[m]] + (
Sqrt[m] (9 (C[2] - C[6]) Sin[(Sqrt[k] t)/Sqrt[m]] +
Sqrt[3] (C[2] - 2 C[4] + C[6]) Sin[(Sqrt[3] Sqrt[k] t)/Sqrt[m]]))/
Sqrt[k]),
1/9 (-3 (C[1] - 2 C[3] + C[5]) Cos[(Sqrt[3] Sqrt[k] t)/Sqrt[
m]] + (-9 f k^(3/2) Cos[t w] +
m w^2 (-3 k +
m w^2) (-3 Sqrt[k] (C[1] + C[3] + C[5] + t (C[2] + C[4] + C[6])) +
Sqrt[3] Sqrt[
m] (C[2] - 2 C[4] + C[6]) Sin[(Sqrt[3] Sqrt[k] t)/Sqrt[m]]) -
9 I f k^(3/2) Sin[t w])/(Sqrt[k] m w^2 (3 k - m w^2))),
1/18 (6 (-((3 E^(I t w) f k^2)/(3 k^2 m w^2 - 4 k m^2 w^4 + m^3 w^6)) + C[
1] + C[3] + C[5] + t (C[2] + C[4] + C[6])) +
9 (-C[1] + C[5]) Cos[(Sqrt[k] t)/Sqrt[m]] +
3 (C[1] - 2 C[3] + C[5]) Cos[(Sqrt[3] Sqrt[k] t)/Sqrt[m]] + (
Sqrt[m] (9 (-C[2] + C[6]) Sin[(Sqrt[k] t)/Sqrt[m]] +
Sqrt[3] (C[2] - 2 C[4] + C[6]) Sin[(Sqrt[3] Sqrt[k] t)/Sqrt[m]]))/
Sqrt[k])}} *)


Making use of Sinc

solt2 = solt /. Sin[x_] :> x*Sinc[x] // Simplify

(* {{1/6 (-((6 E^(I t w) f (k^2 - 3 k m w^2 + m^2 w^4))/(
m w^2 (3 k^2 - 4 k m w^2 + m^2 w^4))) + 2 C[1] + 2 C[3] +
2 C[5] + 2 t (C[2] + C[4] + C[6]) +
3 (C[1] - C[5]) Cos[(Sqrt[k] t)/Sqrt[
m]] + (C[1] - 2 C[3] + C[5]) Cos[(Sqrt[3] Sqrt[k] t)/Sqrt[
m]] + 3 t (C[2] - C[6]) Sinc[(Sqrt[k] t)/Sqrt[m]] +
t (C[2] - 2 C[4] + C[6]) Sinc[(Sqrt[3] Sqrt[k] t)/Sqrt[m]]),
1/9 (-3 (C[1] - 2 C[3] + C[5]) Cos[(Sqrt[3] Sqrt[k] t)/Sqrt[
m]] + (-9 f k^(3/2) Cos[t w] +
m w^2 (-3 k +
m w^2) (-3 Sqrt[
k] (C[1] + C[3] + C[5] + t (C[2] + C[4] + C[6])) +
3 Sqrt[k]
t (C[2] - 2 C[4] + C[6]) Sinc[(Sqrt[3] Sqrt[k] t)/Sqrt[
m]]) - 9 I f k^(3/2) t w Sinc[t w])/(Sqrt[k]
m w^2 (3 k - m w^2))),
1/6 (-((6 E^(I t w) f k^2)/(3 k^2 m w^2 - 4 k m^2 w^4 + m^3 w^6)) +
2 C[1] + 2 C[3] + 2 C[5] + 2 t (C[2] + C[4] + C[6]) +
3 (-C[1] + C[5]) Cos[(Sqrt[k] t)/Sqrt[
m]] + (C[1] - 2 C[3] + C[5]) Cos[(Sqrt[3] Sqrt[k] t)/Sqrt[
m]] + 3 t (-C[2] + C[6]) Sinc[(Sqrt[k] t)/Sqrt[m]] +
t (C[2] - 2 C[4] + C[6]) Sinc[(Sqrt[3] Sqrt[k] t)/Sqrt[m]])}} *)
`
• Thank you so much! Commented Dec 26, 2022 at 23:17