54.10.2 problem 1914

Internal problem ID [13136]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 9, system of higher order odes
Problem number : 1914
Date solved : Sunday, October 12, 2025 at 02:35:47 AM
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*}

✓ Maple. Time used: 0.574 (sec). Leaf size: 92

ode:=[diff(x(t),t) = (a*y(t)+b)*x(t), diff(y(t),t) = (c*x(t)+d)*y(t)]; 
dsolve(ode);
 

\begin{align*} [\{x \left (t \right ) = 0\}, \{y \left (t \right ) &= c_1 \,{\mathrm e}^{d t}\}] \\ \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 {-x \left (t \right ) b +\frac {d}{d t}x \left (t \right )}{x \left (t \right ) a}\right \}\right ] \\ \end{align*}

✓ Mathematica. Time used: 0.151 (sec). Leaf size: 201

ode={D[x[t],t]==(a*y[t]+b)*x[t],D[y[t],t]==(c*x[t]+d)*y[t]}; 
ic={}; 
DSolve[{ode,ic},{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*}

✗ Sympy

from sympy import * 
t = symbols("t") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
d = symbols("d") 
x = Function("x") 
y = Function("y") 
ode=[Eq((-a*y(t) - b)*x(t) + Derivative(x(t), t),0),Eq((-c*x(t) - d)*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

Timed Out