Internal problem ID [10351]
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: 11.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-\left (a \,x^{2 n}+b \,x^{-1+n}\right ) y^{2}=c} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 8014
dsolve(diff(y(x),x)=(a*x^(2*n)+b*x^(n-1))*y(x)^2+c,y(x), singsol=all)
\[ \text {Expression too large to display} \]
✓ Solution by Mathematica
Time used: 2.128 (sec). Leaf size: 1384
DSolve[y'[x]==(a*x^(2*n)+b*x^(n-1))*y[x]^2+c,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {\sqrt {c} (n+1)^2 x^{-n} \left (L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )}{\sqrt {a} c_1 (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {a} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )} y(x)\to \frac {\sqrt {c} (n+1)^2 x^{-n} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\sqrt {a} (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )} y(x)\to \frac {\sqrt {c} (n+1)^2 x^{-n} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\sqrt {a} (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\left (\sqrt {a} \sqrt {-(n+1)^2} n+b \sqrt {c} (n+1)\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {c} b}{\sqrt {a} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )} y(x)\to \frac {\sqrt {c} (n+1) x^{-n} L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\sqrt {a} \sqrt {-(n+1)^2} \left (2 L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}-1}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+L_{-\frac {\sqrt {c} b}{2 \sqrt {a} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {c} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )} \end{align*}