Internal problem ID [9648]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev. Second
edition
Section: Chapter 1, section 1.2. Riccati Equation. 1.2.2. Equations Containing Power Functions
Problem number: 65.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{3} y^{\prime }-a \,x^{3} y^{2}-\left (b \,x^{2}+c \right ) y-s x=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 327
dsolve(x^3*diff(y(x),x)=a*x^3*y(x)^2+(b*x^2+c)*y(x)+s*x,y(x), singsol=all)
\[ y \left (x \right ) = -\frac {2 c_{1} \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{x a \left (\operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) c_{1} +\operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )}+\frac {\left (-1+\sqrt {-4 a s +b^{2}+2 b +1}-b \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2 x a \left (\operatorname {KummerU}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) c_{1} +\operatorname {KummerM}\left (\frac {5}{4}+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.828 (sec). Leaf size: 756
DSolve[x^3*y'[x]==a*x^3*y[x]^2+(b*x^2+c)*y[x]+s*x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {-\left (\sqrt {(b+1)^2-4 a s}-b-1\right ) c^{\frac {1}{2} \sqrt {(b+1)^2-4 a s}} \left (\frac {1}{x}\right )^{\sqrt {(b+1)^2-4 a s}} \left (2 x^2 \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b+\sqrt {(b+1)^2-4 a s}-1\right ),\frac {1}{2} \left (\sqrt {(b+1)^2-4 a s}+2\right ),-\frac {c}{2 x^2}\right )-\frac {2 c \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b+\sqrt {(b+1)^2-4 a s}+3\right ),\frac {1}{2} \left (\sqrt {(b+1)^2-4 a s}+4\right ),-\frac {c}{2 x^2}\right )}{\sqrt {(b+1)^2-4 a s}+2}\right )+c_1 2^{\frac {1}{2} \sqrt {(b+1)^2-4 a s}} \left (\sqrt {(b+1)^2-4 a s}+b+1\right ) \left (2 x^2 \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {(b+1)^2-4 a s}-1\right ),1-\frac {1}{2} \sqrt {(b+1)^2-4 a s},-\frac {c}{2 x^2}\right )+\frac {2 c \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {(b+1)^2-4 a s}+3\right ),2-\frac {1}{2} \sqrt {(b+1)^2-4 a s},-\frac {c}{2 x^2}\right )}{\sqrt {(b+1)^2-4 a s}-2}\right )}{4 a x^3 \left (c^{\frac {1}{2} \sqrt {(b+1)^2-4 a s}} \left (\frac {1}{x}\right )^{\sqrt {(b+1)^2-4 a s}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b+\sqrt {(b+1)^2-4 a s}-1\right ),\frac {1}{2} \left (\sqrt {(b+1)^2-4 a s}+2\right ),-\frac {c}{2 x^2}\right )+c_1 2^{\frac {1}{2} \sqrt {(b+1)^2-4 a s}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {(b+1)^2-4 a s}-1\right ),1-\frac {1}{2} \sqrt {(b+1)^2-4 a s},-\frac {c}{2 x^2}\right )\right )} \\ y(x)\to -\frac {\left (\sqrt {(b+1)^2-4 a s}+b+1\right ) \left (2 x^2 \, _1\tilde {F}_1\left (\frac {1}{4} \left (-b-\sqrt {(b+1)^2-4 a s}-1\right );1-\frac {1}{2} \sqrt {(b+1)^2-4 a s};-\frac {c}{2 x^2}\right )-c \, _1\tilde {F}_1\left (\frac {1}{4} \left (-b-\sqrt {(b+1)^2-4 a s}+3\right );2-\frac {1}{2} \sqrt {(b+1)^2-4 a s};-\frac {c}{2 x^2}\right )\right )}{4 a x^3 \, _1\tilde {F}_1\left (\frac {1}{4} \left (-b-\sqrt {(b+1)^2-4 a s}-1\right );1-\frac {1}{2} \sqrt {(b+1)^2-4 a s};-\frac {c}{2 x^2}\right )} \\ \end{align*}