60.9.56 problem 1912

Internal problem ID [11835]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 8, system of first order odes
Problem number : 1912
Date solved : Sunday, March 30, 2025 at 09:18:14 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=a x_{2} \left (t \right )+b x_{3} \left (t \right ) \cos \left (c t \right )+b x_{4} \left (t \right ) \sin \left (c t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=-a x_{1} \left (t \right )+b x_{3} \left (t \right ) \sin \left (c t \right )-b x_{4} \left (t \right ) \cos \left (c t \right )\\ \frac {d}{d t}x_{3} \left (t \right )&=-b x_{1} \left (t \right ) \cos \left (c t \right )-b x_{2} \left (t \right ) \sin \left (c t \right )+a x_{4} \left (t \right )\\ \frac {d}{d t}x_{4} \left (t \right )&=-b x_{1} \left (t \right ) \sin \left (c t \right )+b x_{2} \left (t \right ) \cos \left (c t \right )-a x_{3} \left (t \right ) \end{align*}

✓ Maple. Time used: 1.119 (sec). Leaf size: 10630

ode:=[diff(x__1(t),t) = a*x__2(t)+b*x__3(t)*cos(c*t)+b*x__4(t)*sin(c*t), diff(x__2(t),t) = -a*x__1(t)+b*x__3(t)*sin(c*t)-b*x__4(t)*cos(c*t), diff(x__3(t),t) = -b*x__1(t)*cos(c*t)-b*x__2(t)*sin(c*t)+a*x__4(t), diff(x__4(t),t) = -b*x__1(t)*sin(c*t)+b*x__2(t)*cos(c*t)-a*x__3(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= c_2 +c_3 \sin \left (c t \right )+c_4 \cos \left (c t \right ) \\ x_{1} \left (t \right ) &= c_1 \,{\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {c^{2} \left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right )}}\, t}{2}}+c_2 \,{\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}-2 \sqrt {c^{2} \left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right )}}\, t}{2}}+c_3 \,{\mathrm e}^{-\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {c^{2} \left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right )}}\, t}{2}}+c_4 \,{\mathrm e}^{\frac {\sqrt {-4 a^{2}-4 a c -4 b^{2}-2 c^{2}+2 \sqrt {c^{2} \left (4 a^{2}+4 a c +4 b^{2}+c^{2}\right )}}\, t}{2}} \\ \text {Expression too large to display} \\ x_{2} \left (t \right ) &= -\cos \left (c t \right ) c_3 +\sin \left (c t \right ) c_4 +c_1 \\ x_{3} \left (t \right ) &= \frac {b \left (\cos \left (c t \right ) c_1 a -\sin \left (c t \right ) c_2 a -c_3 a -c_3 c \right )}{\left (a +c \right ) a} \\ \text {Expression too large to display} \\ x_{4} \left (t \right ) &= \frac {b \left (\cos \left (c t \right ) c_2 a +\sin \left (c t \right ) c_1 a +c_4 a +c_4 c \right )}{\left (a +c \right ) a} \\ \text {Expression too large to display} \\ \end{align*}

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 782

ode={D[ x1[t],t]==a*x2[t]+b*x3[t]*Cos[c*t]+b*x4[t]*Sin[c*t],D[ x2[t],t]==-a*x1[t]+b*x3[t]*Sin[c*t]-b*x4[t]*Cos[c*t],D[ x3[t],t]==-b*x1[t]*Cos[c*t]-b*x2[t]*Sin[c*t]+a*x4[t],D[ x4[t],t]==-b*x1[t]*Sin[c*t]+b*x2[t]*Cos[c*t]-a*x3[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
x__3 = Function("x__3") 
x__4 = Function("x__4") 
ode=[Eq(-a*x__2(t) - b*x__3(t)*cos(c*t) - b*x__4(t)*sin(c*t) + Derivative(x__1(t), t),0),Eq(a*x__1(t) - b*x__3(t)*sin(c*t) + b*x__4(t)*cos(c*t) + Derivative(x__2(t), t),0),Eq(-a*x__4(t) + b*x__1(t)*cos(c*t) + b*x__2(t)*sin(c*t) + Derivative(x__3(t), t),0),Eq(a*x__3(t) + b*x__1(t)*sin(c*t) - b*x__2(t)*cos(c*t) + Derivative(x__4(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t),x__3(t),x__4(t)],ics=ics)
 

NotImplementedError :