8.28.1 problem 5

Internal problem ID [2793]
Book : Differential equations and their applications, 4th ed., M. Braun
Section : Chapter 4. Qualitative theory of differential equations. Section 4.1 (Introduction). Page 377
Problem number : 5
Date solved : Sunday, October 12, 2025 at 01:17:45 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )&=x \left (t \right ) y \left (t \right )^{2}-x \left (t \right )\\ \frac {d}{d t}y \left (t \right )&=x \left (t \right ) \sin \left (\pi y \left (t \right )\right ) \end{align*}

✓ Maple. Time used: 0.999 (sec). Leaf size: 200

ode:=[diff(x(t),t) = x(t)*y(t)^2-x(t), diff(y(t),t) = x(t)*sin(Pi*y(t))]; 
dsolve(ode);
 

\begin{align*} \left \{y \left (t \right ) &= c_1, y \left (t \right ) = \operatorname {RootOf}\left (-\pi ^{3} \int _{}^{\textit {\_Z}}\frac {\operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \csc \left (\pi \textit {\_f} \right )}{\ln \left (1-{\mathrm e}^{i \pi \textit {\_f}}\right ) \operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \pi ^{2} \textit {\_f}^{2}-\ln \left ({\mathrm e}^{i \pi \textit {\_f}}+1\right ) \operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \pi ^{2} \textit {\_f}^{2}-2 i \operatorname {polylog}\left (2, {\mathrm e}^{i \pi \textit {\_f}}\right ) \operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \pi \textit {\_f} +2 i \operatorname {polylog}\left (2, -{\mathrm e}^{i \pi \textit {\_f}}\right ) \operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \pi \textit {\_f} -\operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \pi ^{2} \ln \left (1-{\mathrm e}^{i \pi \textit {\_f}}\right )+\operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \pi ^{2} \ln \left ({\mathrm e}^{i \pi \textit {\_f}}+1\right )+2 \operatorname {polylog}\left (3, {\mathrm e}^{i \pi \textit {\_f}}\right ) \operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right )-2 \operatorname {polylog}\left (3, -{\mathrm e}^{i \pi \textit {\_f}}\right ) \operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right )+c_1 \,\pi ^{3}}d \textit {\_f} +t +c_2 \right )\right \} \\ \left \{x \left (t \right ) &= \frac {\frac {d}{d t}y \left (t \right )}{\sin \left (\pi y \left (t \right )\right )}\right \} \\ \end{align*}

✓ Mathematica. Time used: 0.217 (sec). Leaf size: 403

ode={D[x[t],t]==x[t]*y[t]^2-x[t],D[y[t],t]==x[t]*Sin[Pi*y[t]]}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)&\to \text {InverseFunction}\left [\frac {\text {arctanh}(\cos (\pi \text {$\#$1}))}{\pi }+\frac {\pi ^2 \text {$\#$1}^2 \left (\log \left (1-e^{i \pi \text {$\#$1}}\right )-\log \left (1+e^{i \pi \text {$\#$1}}\right )\right )+2 i \pi \text {$\#$1} \left (\operatorname {PolyLog}\left (2,-e^{i \pi \text {$\#$1}}\right )-\operatorname {PolyLog}\left (2,e^{i \pi \text {$\#$1}}\right )\right )+2 \left (\operatorname {PolyLog}\left (3,e^{i \pi \text {$\#$1}}\right )-\operatorname {PolyLog}\left (3,-e^{i \pi \text {$\#$1}}\right )\right )}{\pi ^3}\&\right ][x(t)+c_1]\\ \text {Solve}\left [\int _1^{x(t)}\frac {1}{K[1] \left (\text {InverseFunction}\left [\frac {\text {arctanh}(\cos (\pi \text {$\#$1}))}{\pi }+\frac {\pi ^2 \left (\log \left (1-e^{i \pi \text {$\#$1}}\right )-\log \left (1+e^{i \pi \text {$\#$1}}\right )\right ) \text {$\#$1}^2+2 i \pi \left (\operatorname {PolyLog}\left (2,-e^{i \pi \text {$\#$1}}\right )-\operatorname {PolyLog}\left (2,e^{i \pi \text {$\#$1}}\right )\right ) \text {$\#$1}+2 \left (\operatorname {PolyLog}\left (3,e^{i \pi \text {$\#$1}}\right )-\operatorname {PolyLog}\left (3,-e^{i \pi \text {$\#$1}}\right )\right )}{\pi ^3}\&\right ][c_1+K[1]]-1\right ) \left (\text {InverseFunction}\left [\frac {\text {arctanh}(\cos (\pi \text {$\#$1}))}{\pi }+\frac {\pi ^2 \left (\log \left (1-e^{i \pi \text {$\#$1}}\right )-\log \left (1+e^{i \pi \text {$\#$1}}\right )\right ) \text {$\#$1}^2+2 i \pi \left (\operatorname {PolyLog}\left (2,-e^{i \pi \text {$\#$1}}\right )-\operatorname {PolyLog}\left (2,e^{i \pi \text {$\#$1}}\right )\right ) \text {$\#$1}+2 \left (\operatorname {PolyLog}\left (3,e^{i \pi \text {$\#$1}}\right )-\operatorname {PolyLog}\left (3,-e^{i \pi \text {$\#$1}}\right )\right )}{\pi ^3}\&\right ][c_1+K[1]]+1\right )}dK[1]=t+c_2,x(t)\right ] \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t)*y(t)**2 + x(t) + Derivative(x(t), t),0),Eq(-x(t)*sin(pi*y(t)) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

NotImplementedError : multiple generators [log(4*C1**6/(C1**7 - 11*C1**5 + 19*C1**3 - 9*C1) + 4*C1**4/(C1**7 - 11*C1**5 + 19*C1**3 - 9*C1) + 6*C1**4/(C1**5 - 10*C1**3 + 9*C1) - 20*C1**2/(C1**7 - 11*C1**5 + 19*C1**3 - 9*C1) - 12*C1**2/(C1**5 - 10*C1**3 + 9*C1) - 10*C1**2/(C1**3 - 9*C1) + u + 12/(C1**7 - 11*C1**5 + 19*C1**3 - 9*C1) + 6/(C1**5 - 10*C1**3 + 9*C1) - 6/(C1**3 - 9*C1)), log(C1**6/(C1**5 + 2*C1**4 - 8*C1**3 - 18*C1**2 - 9*C1) + C1**4/(C1**5 + 2*C1**4 - 8*C1**3 - 18*C1**2 - 9*C1) + 3*C1**4/(C1**4 + C1**3 - 9*C1**2 - 9*C1) - 5*C1**2/(C1**5 + 2*C1**4 - 8*C1**3 - 18*C1**2 - 9*C1) - 6*C1**2/(C1**4 + C1**3 - 9*C1**2 - 9*C1) - 10*C1**2/(C1**3 - 9*C1) + u + 3/(C1**5 + 2*C1**4 - 8*C1**3 - 18*C1**2 - 9*C1) + 3/(C1**4 + C1**3 - 9*C1**2 - 9*C1) - 6/(C1**3 - 9*C1)), log(C1**6/(C1**5 - 2*C1**4 - 8*C1**3 + 18*C1**2 - 9*C1) + C1**4/(C1**5 - 2*C1**4 - 8*C1**3 + 18*C1**2 - 9*C1) - 3*C1**4/(C1**4 - C1**3 - 9*C1**2 + 9*C1) - 5*C1**2/(C1**5 - 2*C1**4 - 8*C1**3 + 18*C1**2 - 9*C1) + 6*C1**2/(C1**4 - C1**3 - 9*C1**2 + 9*C1) - 10*C1**2/(C1**3 - 9*C1) + u + 3/(C1**5 - 2*C1**4 - 8*C1**3 + 18*C1**2 - 9*C1) - 3/(C1**4 - C1**3 - 9*C1**2 + 9*C1) - 6/(C1**3 - 9*C1))] 
No algorithms are implemented to solve equation -C2 - t + log(u + (C1**6/(C1 + 1)**2 + 3*C1**4/(C1 + 1) + C1**4/(C1 + 1)**2 - 10*C1**2 - 6*C1**2/(C1 + 1) - 5*C1**2/(C1 + 1)**2 - 6 + 3/(C1 + 1) + 3/(C1 + 1)**2)/(C1**3 - 9*C1))/(2*(C1 + 1)) - log(u + (C1**6/(C1 - 1)**2 - 3*C1**4/(C1 - 1) + C1**4/(C1 - 1)**2 - 10*C1**2 + 6*C1**2/(C1 - 1) - 5*C1**2/(C1 - 1)**2 - 6 - 3/(C1 - 1) + 3/(C1 - 1)**2)/(C1**3 - 9*C1))/(2*(C1 - 1)) + log(u + (4*C1**6/((C1 - 1)**2*(C1 + 1)**2) + 6*C1**4/((C1 - 1)*(C1 + 1)) + 4*C1**4/((C1 - 1)**2*(C1 + 1)**2) - 10*C1**2 - 12*C1**2/((C1 - 1)*(C1 + 1)) - 20*C1**2/((C1 - 1)**2*(C1 + 1)**2) - 6 + 6/((C1 - 1)*(C1 + 1)) + 12/((C1 - 1)**2*(C1 + 1)**2))/(C1**3 - 9*C1))/((C1 - 1)*(C1 + 1))