Internal problem ID [15502]
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.
ODE order: 1.
ODE degree: 1.
Solve \begin {align*} x_{1}^{\prime }\left (t \right )&=\frac {x_{1} \left (t \right )^{2}}{x_{2} \left (t \right )}\\ x_{2}^{\prime }\left (t \right )&=x_{2} \left (t \right )-x_{1} \left (t \right ) \end {align*}
✓ Solution by Maple
Time used: 0.047 (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: 0.102 (sec). Leaf size: 143
DSolve[{x1'[t]==x1[t]^2/x2[t],x2'[t]==x2[t]-x1[t]},{x1[t],x2[t]},t,IncludeSingularSolutions -> True]
\begin{align*} \text {x2}(t)\to 2 i e^{\frac {t}{2}+c_2} \sqrt {-1+2 c_1 e^{t+2 c_2}} \\ \text {x1}(t)\to -\frac {i e^{\frac {t}{2}+c_2}}{\sqrt {-1+2 c_1 e^{t+2 c_2}}} \\ \text {x2}(t)\to -2 i e^{\frac {t}{2}+c_2} \sqrt {-1+2 c_1 e^{t+2 c_2}} \\ \text {x1}(t)\to \frac {i e^{\frac {t}{2}+c_2}}{\sqrt {-1+2 c_1 e^{t+2 c_2}}} \\ \end{align*}