75.26.1 problem 767

Internal problem ID [17229]
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 : Tuesday, January 28, 2025 at 08:27:32 PM
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*}

✓ Solution by Maple

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

dsolve([diff(x__1(t),t)=-2*t*x__1(t)^2,diff(x__2(t),t)=(x__2(t)+t)/t],singsol=all)
 

\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*}

✓ Solution by Mathematica

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

DSolve[{D[ x1[t],t]==-2*t*x1[t]^2,D[ x2[t],t]==(x2[t]+t)/t},{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*}