75.26.1 problem 767

Internal problem ID [17150]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 3 (Systems of differential equations). Section 19. Basic concepts and definitions. Exercises page 199
Problem number : 767
Date solved : Friday, March 14, 2025 at 04:49:20 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-2 t x_{1} \left (t \right )^{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {x_{2} \left (t \right )+t}{t} \end{align*}

✓ Maple. Time used: 0.143 (sec). Leaf size: 22

ode:=[diff(x__1(t),t) = -2*t*x__1(t)^2, diff(x__2(t),t) = (x__2(t)+t)/t]; 
dsolve(ode);
 

\begin{align*} \left \{x_{1} \left (t \right ) &= \frac {1}{t^{2}+c_{2}}\right \} \\ \{x_{2} \left (t \right ) &= \left (\ln \left (t \right )+c_{1} \right ) t\} \\ \end{align*}

✓ Mathematica. Time used: 0.157 (sec). Leaf size: 40

ode={D[ x1[t],t]==-2*t*x1[t]^2,D[ x2[t],t]==(x2[t]+t)/t}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to \frac {1}{t^2-c_1} \\ \text {x2}(t)\to t (\log (t)+c_2) \\ \text {x1}(t)\to 0 \\ \text {x2}(t)\to t (\log (t)+c_2) \\ \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(2*t*x__1(t)**2 + Derivative(x__1(t), t),0),Eq(Derivative(x__2(t), t) - (t + x__2(t))/t,0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

NotImplementedError :