Internal problem ID [10395]
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]
\[ \boxed {x^{3} y^{\prime }-a \,x^{3} y^{2}-\left (b \,x^{2}+c \right ) y=s x} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 435
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)
\[ y \left (x \right ) = -\frac {\left (\frac {\left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{4}+c_{1} \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )\right ) \left (\sqrt {-4 a s +b^{2}+2 b +1}+b +1\right ) x \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right )}{2 a \left (\frac {\left (\left (b -1\right ) x^{2}+c \right ) \left (1-\sqrt {-4 a s +b^{2}+2 b +1}+b \right ) \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+\left (\frac {1}{2}+\frac {\left (b -1\right ) \sqrt {-4 a s +b^{2}+2 b +1}}{2}+a s -\frac {b^{2}}{2}\right ) x^{2} \operatorname {KummerM}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )-4 c_{1} \left (\frac {\left (\left (-b +1\right ) x^{2}-c \right ) \operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}+\frac {1}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right )}{2}+\operatorname {KummerU}\left (\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{4}+\frac {b}{4}-\frac {3}{4}, 1+\frac {\sqrt {-4 a s +b^{2}+2 b +1}}{2}, \frac {c}{2 x^{2}}\right ) x^{2}\right )\right )} \]
✓ Solution by Mathematica
Time used: 3.199 (sec). Leaf size: 907
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 (\left (\sqrt {-4 a s+b^2+2 b+1}-b-1\right ) c^{\frac {1}{2} \sqrt {-4 a s+b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a s+b^2+2 b+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b+\sqrt {b^2+2 b-4 a s+1}-1\right ),\frac {1}{2} \left (\sqrt {b^2+2 b-4 a s+1}+2\right ),-\frac {c}{2 x^2}\right )\right )+\frac {\left (\sqrt {-4 a s+b^2+2 b+1}-b-1\right ) c^{\frac {1}{2} \sqrt {-4 a s+b^2+2 b+1}+1} \left (\frac {1}{x}\right )^{\sqrt {-4 a s+b^2+2 b+1}+2} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b+\sqrt {b^2+2 b-4 a s+1}+3\right ),\frac {1}{2} \left (\sqrt {b^2+2 b-4 a s+1}+4\right ),-\frac {c}{2 x^2}\right )}{\sqrt {-4 a s+b^2+2 b+1}+2}+\frac {c_1 2^{\frac {1}{2} \sqrt {-4 a s+b^2+2 b+1}} \left (\sqrt {-4 a s+b^2+2 b+1}+b+1\right ) \left (x^2 \left (\sqrt {-4 a s+b^2+2 b+1}-2\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+2 b-4 a s+1}-1\right ),1-\frac {1}{2} \sqrt {b^2+2 b-4 a s+1},-\frac {c}{2 x^2}\right )+c \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+2 b-4 a s+1}+3\right ),2-\frac {1}{2} \sqrt {b^2+2 b-4 a s+1},-\frac {c}{2 x^2}\right )\right )}{x^2 \left (\sqrt {-4 a s+b^2+2 b+1}-2\right )}}{2 a x \left (c^{\frac {1}{2} \sqrt {-4 a s+b^2+2 b+1}} \left (\frac {1}{x}\right )^{\sqrt {-4 a s+b^2+2 b+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b+\sqrt {b^2+2 b-4 a s+1}-1\right ),\frac {1}{2} \left (\sqrt {b^2+2 b-4 a s+1}+2\right ),-\frac {c}{2 x^2}\right )+c_1 2^{\frac {1}{2} \sqrt {-4 a s+b^2+2 b+1}} \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+2 b-4 a s+1}-1\right ),1-\frac {1}{2} \sqrt {b^2+2 b-4 a s+1},-\frac {c}{2 x^2}\right )\right )} \\ y(x)\to \frac {\frac {c \left (b \left (\sqrt {-4 a s+b^2+2 b+1}+4\right )+3 \sqrt {-4 a s+b^2+2 b+1}-4 a s+b^2+3\right ) \operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+2 b-4 a s+1}+3\right ),2-\frac {1}{2} \sqrt {b^2+2 b-4 a s+1},-\frac {c}{2 x^2}\right )}{\operatorname {Hypergeometric1F1}\left (\frac {1}{4} \left (-b-\sqrt {b^2+2 b-4 a s+1}-1\right ),1-\frac {1}{2} \sqrt {b^2+2 b-4 a s+1},-\frac {c}{2 x^2}\right )}+x^2 \left (-4 a s+b^2+2 b-3\right ) \left (\sqrt {-4 a s+b^2+2 b+1}+b+1\right )}{2 a x^3 \left (4 a s-b^2-2 b+3\right )} \\ \end{align*}