60.9.33 problem 1888
Internal
problem
ID
[11887]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
8,
system
of
first
order
odes
Problem
number
:
1888
Date
solved
:
Tuesday, January 28, 2025 at 06:24:00 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=a_{1} x \left (t \right )+b_{1} y \left (t \right )+c_{1}\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=a_{2} x \left (t \right )+b_{2} y \left (t \right )+c_{2} \end{align*}
✓ Solution by Maple
Time used: 0.211 (sec). Leaf size: 646
dsolve([diff(x(t),t,t)=a__1*x(t)+b__1*y(t)+c__1,diff(y(t),t,t)=a__2*x(t)+b__2*y(t)+c__2],singsol=all)
\begin{align*}
x \left (t \right ) &= -\frac {-b_{1} c_{2} +b_{2} c_{1}}{a_{1} b_{2} -a_{2} b_{1}}+c_3 \,{\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}+c_4 \,{\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}+c_5 \,{\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} +2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}+c_6 \,{\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} +2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}} \\
y \left (t \right ) &= -\frac {\left (-c_6 a_{1} b_{2}^{2}+\left (-a_{1} \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_6 +a_{1}^{2} c_6 +c_6 a_{2} b_{1} \right ) b_{2} +\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_6 -a_{1} c_6 \right ) a_{2} b_{1} \right ) {\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} +2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (-c_5 a_{1} b_{2}^{2}+\left (-a_{1} \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_5 +a_{1}^{2} c_5 +c_5 a_{2} b_{1} \right ) b_{2} +\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_5 -a_{1} c_5 \right ) a_{2} b_{1} \right ) {\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} +2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (-c_4 a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_4 a_{1} +c_4 \,a_{1}^{2}+c_4 a_{2} b_{1} \right ) b_{2} +\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_4 -c_4 a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {\left (-c_3 a_{1} b_{2}^{2}+\left (\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_3 a_{1} +c_3 \,a_{1}^{2}+c_3 a_{2} b_{1} \right ) b_{2} +\left (-\sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}\, c_3 -c_3 a_{1} \right ) a_{2} b_{1} \right ) {\mathrm e}^{-\frac {\sqrt {2 a_{1} +2 b_{2} -2 \sqrt {a_{1}^{2}-2 a_{1} b_{2} +4 a_{2} b_{1} +b_{2}^{2}}}\, t}{2}}}{2 b_{1} \left (a_{1} b_{2} -a_{2} b_{1} \right )}-\frac {2 a_{1} c_{2} -2 a_{2} c_{1}}{2 \left (a_{1} b_{2} -a_{2} b_{1} \right )} \\
\end{align*}
✓ Solution by Mathematica
Time used: 19.813 (sec). Leaf size: 19318
DSolve[{D[x[t],{t,2}]==a1*x[t]+b1*y[t]+c1,D[y[t],{t,2}]==a2*x[t]+b2*y[t]+c2},{x[t],y[t]},t,IncludeSingularSolutions -> True]
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