60.1.178 problem 179

Internal problem ID [10192]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 179
Date solved : Monday, January 27, 2025 at 06:36:21 PM
CAS classification : [_rational, _Riccati]

\begin{align*} 3 x \left (x^{2}-1\right ) y^{\prime }+x y^{2}-\left (x^{2}+1\right ) y-3 x&=0 \end{align*}

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 190

dsolve(3*x*(x^2-1)*diff(y(x),x) + x*y(x)^2 - (x^2+1)*y(x) - 3*x=0,y(x), singsol=all)
 

\[ y = \frac {80 \left (x^{2}-\frac {2}{5}\right ) \pi c_{1} \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right )+315 \left (\frac {24 \left (x^{2}\right )^{{1}/{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right ) x^{{4}/{3}}}{35}+\left (x^{2}\right )^{{1}/{6}} \left (-x^{2}+1\right )^{{5}/{6}} \left (\left (x^{4}-x^{2}\right ) c_{1} \operatorname {hypergeom}\left (\left [\frac {11}{6}, \frac {13}{6}\right ], \left [\frac {7}{3}\right ], x^{2}\right )-\frac {6 \operatorname {hypergeom}\left (\left [\frac {3}{2}, \frac {11}{6}\right ], \left [\frac {5}{3}\right ], x^{2}\right ) \left (x^{{4}/{3}}-x^{{10}/{3}}\right )}{7}\right )\right ) \Gamma \left (\frac {2}{3}\right )}{x^{{1}/{3}} \left (16 x^{{2}/{3}} \pi \sqrt {3}\, \operatorname {LegendreP}\left (-\frac {1}{6}, -\frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) c_{1} +72 \left (x^{2}\right )^{{1}/{3}} \operatorname {LegendreP}\left (-\frac {1}{6}, \frac {1}{3}, \frac {-x^{2}-1}{x^{2}-1}\right ) \Gamma \left (\frac {2}{3}\right )^{2}\right )} \]

✓ Solution by Mathematica

Time used: 3.152 (sec). Leaf size: 4443

DSolve[3*x*(x^2-1)*D[y[x],x] + x*y[x]^2 - (x^2+1)*y[x] - 3*x==0,y[x],x,IncludeSingularSolutions -> True]
 

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