Internal problem ID [9487]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 8, system of first order odes
Problem number: 1912.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\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 )\\ x_{2}^{\prime }\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 )\\ x_{3}^{\prime }\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 )\\ x_{4}^{\prime }\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*}
✓ Solution by Maple
Time used: 0.953 (sec). Leaf size: 10632
dsolve([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)],[x__1(t), x__2(t), x__3(t), x__4(t)], singsol=all)
\begin{align*} x_{1} \left (t \right ) = -\frac {b \left (\cos \left (t c \right ) c_{2} a -\sin \left (t c \right ) c_{1} a +c_{4} a +c_{4} c \right )}{\left (a +c \right ) a} \\ x_{2} \left (t \right ) = -\frac {b \left (\cos \left (t c \right ) c_{1} a +\sin \left (t c \right ) c_{2} a +c_{3} a +c_{3} c \right )}{\left (a +c \right ) a} \\ x_{3} \left (t \right ) = \cos \left (t c \right ) c_{3} -\sin \left (t c \right ) c_{4} +c_{1} \\ x_{4} \left (t \right ) = c_{2} +c_{3} \sin \left (t c \right )+c_{4} \cos \left (t c \right ) \\ \end{align*} \begin{align*} \text {Expression too large to display} \\ \text {Expression too large to display} \\ \text {Expression too large to display} \\ x_{4} \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}} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 700
DSolve[{x1'[t]==a*x2[t]+b*x3[t]*Cos[c*t]+b*x4[t]*Sin[c*t],x2'[t]==-a*x1[t]+b*x3[t]*Sin[c*t]-b*x4[t]*Cos[c*t],x3'[t]==-b*x1[t]*Cos[c*t]-b*x2[t]*Sin[c*t]+a*x4[t],x4'[t]==-b*x1[t]*Sin[c*t]+b*x2[t]*Cos[c*t]-a*x3[t]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x1}(t)\to c_3 \cos \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )+c_1 \cos \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right )+c_4 \sin \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )+c_2 \sin \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right ) \\ \text {x2}(t)\to -c_4 \cos \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )-c_2 \cos \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right )+c_3 \sin \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )+c_1 \sin \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right ) \\ \text {x3}(t)\to \frac {\frac {1}{2} c_2 \left (\sqrt {(2 a+c)^2+4 b^2}+2 a+c\right ) \cos \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )+\frac {1}{2} c_4 \left (-\sqrt {(2 a+c)^2+4 b^2}+2 a+c\right ) \cos \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right )+\frac {1}{2} c_1 \left (\sqrt {(2 a+c)^2+4 b^2}+2 a+c\right ) \sin \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )+\frac {1}{2} c_3 \left (-\sqrt {(2 a+c)^2+4 b^2}+2 a+c\right ) \sin \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right )}{b} \\ \text {x4}(t)\to \frac {-\frac {1}{2} c_1 \left (\sqrt {(2 a+c)^2+4 b^2}+2 a+c\right ) \cos \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )+\frac {1}{2} c_3 \left (\sqrt {(2 a+c)^2+4 b^2}-2 a-c\right ) \cos \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right )+\frac {1}{2} c_2 \left (\sqrt {(2 a+c)^2+4 b^2}+2 a+c\right ) \sin \left (\frac {1}{2} t \left (c-\sqrt {(2 a+c)^2+4 b^2}\right )\right )+\frac {1}{2} c_4 \left (-\sqrt {(2 a+c)^2+4 b^2}+2 a+c\right ) \sin \left (\frac {1}{2} t \left (\sqrt {(2 a+c)^2+4 b^2}+c\right )\right )}{b} \\ \end{align*}