Internal problem ID [10382]
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: 52.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
\[ \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-y b x=c \,x^{2 n}+s \,x^{n}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 373
dsolve(x^2*diff(y(x),x)=a*x^2*y(x)^2+b*x*y(x)+c*x^(2*n)+s*x^n,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\operatorname {KummerM}\left (\frac {\left (b -n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) \left (\left (-b -n -1\right ) \sqrt {c}+i \sqrt {a}\, s \right )+2 \sqrt {c}\, \operatorname {KummerU}\left (\frac {\left (b -n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_{1} n -2 \left (\operatorname {KummerU}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_{1} +\operatorname {KummerM}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right )\right ) \left (\frac {\left (b -n +1\right ) \sqrt {c}}{2}+i \sqrt {a}\, \left (c \,x^{n}+\frac {s}{2}\right )\right )}{2 \sqrt {c}\, x a \left (\operatorname {KummerU}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right ) c_{1} +\operatorname {KummerM}\left (\frac {\left (b +n +1\right ) \sqrt {c}+i \sqrt {a}\, s}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {a}\, \sqrt {c}\, x^{n}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.839 (sec). Leaf size: 819
DSolve[x^2*y'[x]==a*x^2*y[x]^2+b*x*y[x]+c*x^(2*n)+s*x^n,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {i \sqrt {a} c_1 x^n \left (\sqrt {c} (b+n+1)-i \sqrt {a} s\right ) \operatorname {HypergeometricU}\left (\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+2 n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+c_1 n \left (i \sqrt {a} \sqrt {c} x^n+b+1\right ) \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+n \left (2 i \sqrt {a} \sqrt {c} x^n L_{-\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n}}^{\frac {b+n+1}{n}}\left (-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+\left (i \sqrt {a} \sqrt {c} x^n+b+1\right ) L_{-\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n}}^{\frac {b+1}{n}}\left (-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )\right )}{a n x \left (c_1 \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )+L_{-\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n}}^{\frac {b+1}{n}}\left (-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )\right )} \\ y(x)\to -\frac {\frac {\sqrt {a} x^n \left (\sqrt {a} s+i \sqrt {c} (b+n+1)\right ) \operatorname {HypergeometricU}\left (\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+2 n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}{n \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}+i \sqrt {a} \sqrt {c} x^n+b+1}{a x} \\ y(x)\to -\frac {\frac {\sqrt {a} x^n \left (\sqrt {a} s+i \sqrt {c} (b+n+1)\right ) \operatorname {HypergeometricU}\left (\frac {b+3 n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+2 n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}{n \operatorname {HypergeometricU}\left (\frac {b+n-\frac {i \sqrt {a} s}{\sqrt {c}}+1}{2 n},\frac {b+n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}+i \sqrt {a} \sqrt {c} x^n+b+1}{a x} \\ \end{align*}