60.10.2 problem 1914
Internal
problem
ID
[11913]
Book
:
Differential
Gleichungen,
E.
Kamke,
3rd
ed.
Chelsea
Pub.
NY,
1948
Section
:
Chapter
9,
system
of
higher
order
odes
Problem
number
:
1914
Date
solved
:
Tuesday, January 28, 2025 at 06:24:06 PM
CAS
classification
:
system_of_ODEs
\begin{align*} \frac {d}{d t}x \left (t \right )&=\left (a y \left (t \right )+b \right ) x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=\left (c x \left (t \right )+d \right ) y \left (t \right ) \end{align*}
✓ Solution by Maple
Time used: 1.109 (sec). Leaf size: 92
dsolve([diff(x(t),t)=(a*y(t)+b)*x(t),diff(y(t),t)=(c*x(t)+d)*y(t)],singsol=all)
\begin{align*}
\\
\left [\left \{x \left (t \right ) &= \operatorname {RootOf}\left (-\int _{}^{\textit {\_Z}}\frac {1}{b \textit {\_a} \left (\operatorname {LambertW}\left (\frac {{\mathrm e}^{-1} \textit {\_a}^{\frac {d}{b}} {\mathrm e}^{\frac {\textit {\_a} c}{b}} {\mathrm e}^{\frac {c_{1}}{b}}}{b}\right )+1\right )}d \textit {\_a} +t +c_{2} \right )\right \}, \left \{y \left (t \right ) = \frac {-b x \left (t \right )+\frac {d}{d t}x \left (t \right )}{a x \left (t \right )}\right \}\right ] \\
\end{align*}
✓ Solution by Mathematica
Time used: 0.224 (sec). Leaf size: 201
DSolve[{D[x[t],t]==(a*y[t]+b)*x[t],D[y[t],t]==(c*x[t]+d)*y[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
\begin{align*}
y(t)\to \frac {b W\left (\frac {a \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2]{}^{\frac {d}{b}} \exp \left (\frac {c \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2]+c_1}{b}\right )}{b}\right )}{a} \\
x(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (W\left (\frac {a e^{\frac {c_1}{b}+\frac {c K[1]}{b}} K[1]^{\frac {d}{b}}}{b}\right )+1\right )}dK[1]\&\right ][b t+c_2] \\
\end{align*}