60.9.57 problem 1912
Internal
problem
ID
[11911]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1912
Date
solved
:
Tuesday, January 28, 2025 at 06:24:05 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*}
✓ Solution by Maple
Time used: 0.977 (sec). Leaf size: 10630
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)],singsol=all)
\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*}
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 782
DSolve[{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]},{x1[t],x2[t],x3[t],x4[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
\text {x1}(t)\to c_1 \cos \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )+c_2 \sin \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )+c_3 \cos \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right )+c_4 \sin \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right ) \\
\text {x2}(t)\to -c_2 \cos \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )+c_1 \sin \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )-c_4 \cos \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right )+c_3 \sin \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right ) \\
\text {x3}(t)\to \frac {c_4 \left (-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \cos \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )+c_3 \left (-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \sin \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )+c_2 \left (\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \cos \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right )+c_1 \left (\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \sin \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right )}{b} \\
\text {x4}(t)\to \frac {-\left (c_3 \left (-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \cos \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )\right )+c_4 \left (-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \sin \left (\frac {1}{2} t \left (\sqrt {4 a^2+4 a c+4 b^2+c^2}+c\right )\right )-c_1 \left (\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \cos \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right )+c_2 \left (\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}+a+\frac {c}{2}\right ) \sin \left (t \left (\frac {c}{2}-\frac {1}{2} \sqrt {(2 a+c)^2+4 b^2}\right )\right )}{b} \\
\end{align*}