14.28.1 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 : Tuesday, January 28, 2025 at 02:38:53 PM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 2.298 (sec). Leaf size: 299

dsolve([diff(x(t),t)=x(t)*y(t)^2-x(t),diff(y(t),t)=x(t)*sin(Pi*y(t))],singsol=all)
 

\begin{align*} \left \{y &= c_1, y = \operatorname {RootOf}\left (-\pi ^{3} \left (\int _{}^{\textit {\_Z}}\frac {\operatorname {csgn}\left (\csc \left (\pi \textit {\_f} \right )\right ) \csc \left (\pi \textit {\_f} \right )}{2 \sqrt {-\frac {{\mathrm e}^{2 i \pi \textit {\_f}}}{\left ({\mathrm e}^{2 i \pi \textit {\_f}}-1\right )^{2}}}\, \ln \left (1-{\mathrm e}^{i \pi \textit {\_f}}\right ) \sin \left (\pi \textit {\_f} \right ) \pi ^{2} \textit {\_f}^{2}-2 \sqrt {-\frac {{\mathrm e}^{2 i \pi \textit {\_f}}}{\left ({\mathrm e}^{2 i \pi \textit {\_f}}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{i \pi \textit {\_f}}+1\right ) \sin \left (\pi \textit {\_f} \right ) \pi ^{2} \textit {\_f}^{2}-4 i \sqrt {-\frac {{\mathrm e}^{2 i \pi \textit {\_f}}}{\left ({\mathrm e}^{2 i \pi \textit {\_f}}-1\right )^{2}}}\, \operatorname {polylog}\left (2, {\mathrm e}^{i \pi \textit {\_f}}\right ) \sin \left (\pi \textit {\_f} \right ) \pi \textit {\_f} +4 i \sqrt {-\frac {{\mathrm e}^{2 i \pi \textit {\_f}}}{\left ({\mathrm e}^{2 i \pi \textit {\_f}}-1\right )^{2}}}\, \operatorname {polylog}\left (2, -{\mathrm e}^{i \pi \textit {\_f}}\right ) \sin \left (\pi \textit {\_f} \right ) \pi \textit {\_f} +4 \sqrt {-\frac {{\mathrm e}^{2 i \pi \textit {\_f}}}{\left ({\mathrm e}^{2 i \pi \textit {\_f}}-1\right )^{2}}}\, \operatorname {arctanh}\left ({\mathrm e}^{i \pi \textit {\_f}}\right ) \sin \left (\pi \textit {\_f} \right ) \pi ^{2}+c_1 \,\pi ^{3}+4 \sqrt {-\frac {{\mathrm e}^{2 i \pi \textit {\_f}}}{\left ({\mathrm e}^{2 i \pi \textit {\_f}}-1\right )^{2}}}\, \operatorname {polylog}\left (3, {\mathrm e}^{i \pi \textit {\_f}}\right ) \sin \left (\pi \textit {\_f} \right )-4 \sqrt {-\frac {{\mathrm e}^{2 i \pi \textit {\_f}}}{\left ({\mathrm e}^{2 i \pi \textit {\_f}}-1\right )^{2}}}\, \operatorname {polylog}\left (3, -{\mathrm e}^{i \pi \textit {\_f}}\right ) \sin \left (\pi \textit {\_f} \right )}d \textit {\_f} \right )+t +c_2 \right )\right \} \\ \left \{x \left (t \right ) &= \frac {y^{\prime }}{\sin \left (\pi y\right )}\right \} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.338 (sec). Leaf size: 403

DSolve[{D[x[t],t]==x[t]*y[t]^2-x[t],D[y[t],t]==x[t]*Sin[Pi*y[t]]},{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*}