90.33.1 problem 1

Internal problem ID [25475]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 9. Linear Systems of Differential Equations. Exercises at page 645
Problem number : 1
Date solved : Sunday, October 12, 2025 at 05:55:39 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=y_{1} \left (t \right ) y_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 32

ode:=[diff(y__1(t),t) = y__2(t), diff(y__2(t),t) = y__1(t)*y__2(t)]; 
dsolve(ode);
 

\begin{align*} \left \{y_{1} \left (t \right ) &= \frac {\tan \left (\frac {\left (t +c_2 \right ) \sqrt {2}}{2 c_1}\right ) \sqrt {2}}{c_1}\right \} \\ \{y_{2} \left (t \right ) &= \frac {d}{d t}y_{1} \left (t \right )\} \\ \end{align*}

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 63

ode={D[y1[t],t]==y2[t],D[y2[t],t]==y1[t]*y2[t]}; 
ic={}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y2}(t)&\to c_1 \sec ^2\left (\frac {\sqrt {c_1} (t+2 c_2)}{\sqrt {2}}\right )\\ \text {y1}(t)&\to \sqrt {2} \sqrt {c_1} \tan \left (\frac {\sqrt {c_1} (t+2 c_2)}{\sqrt {2}}\right ) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-y2(t) + Derivative(y1(t), t),0),Eq(-y1(t)*y2(t) + Derivative(y2(t), t),0)] 
ics = {} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

NotImplementedError : multiple generators [log(-sqrt(2)*C1*sqrt(-1/C1) + u), log(sqrt(2)*C1*sqrt(-1/