75.26.4 problem 771

Internal problem ID [17232]
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 : 771
Date solved : Tuesday, January 28, 2025 at 08:27:34 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 66

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

\begin{align*} \\ \left [\left \{x_{1} \left (t \right ) &= \frac {1}{\sqrt {2 \,{\mathrm e}^{-t} c_{1} -2 c_{2}}}, x_{1} \left (t \right ) = -\frac {1}{\sqrt {2 \,{\mathrm e}^{-t} c_{1} -2 c_{2}}}\right \}, \left \{x_{2} \left (t \right ) = \frac {x_{1} \left (t \right )^{2}}{\frac {d}{d t}x_{1} \left (t \right )}\right \}\right ] \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.099 (sec). Leaf size: 125

DSolve[{D[ x1[t],t]==x1[t]^2/x2[t],D[ x2[t],t]==x2[t]-x1[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} \text {x2}(t)\to \frac {2 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (2 c_1 K[1]^2+1\right )}dK[1]\&\right ]\left [\frac {t}{2}+c_2\right ]}{1+2 c_1 \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (2 c_1 K[1]^2+1\right )}dK[1]\&\right ]\left [\frac {t}{2}+c_2\right ]{}^2} \\ \text {x1}(t)\to \text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (2 c_1 K[1]^2+1\right )}dK[1]\&\right ]\left [\frac {t}{2}+c_2\right ] \\ \end{align*}