Internal problem ID [9639]
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]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-a \,x^{2} y^{2}-b x y-c \,x^{2 n}-s \,x^{n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 442
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)
\[ y \relax (x ) = -\frac {\left (2 i \sqrt {a}\, x^{n} c_{1} c +i \sqrt {a}\, c_{1} s +\sqrt {c}\, c_{1} b -\sqrt {c}\, c_{1} n +\sqrt {c}\, c_{1}\right ) \KummerU \left (\frac {i \sqrt {a}\, s +\sqrt {c}\, b +\sqrt {c}\, n +\sqrt {c}}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right )-2 \sqrt {c}\, \KummerU \left (\frac {i \sqrt {a}\, s +\sqrt {c}\, b -\sqrt {c}\, n +\sqrt {c}}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right ) c_{1} n +\left (2 i \sqrt {a}\, x^{n} c +i \sqrt {a}\, s +\sqrt {c}\, b -\sqrt {c}\, n +\sqrt {c}\right ) \KummerM \left (\frac {i \sqrt {a}\, s +\sqrt {c}\, b +\sqrt {c}\, n +\sqrt {c}}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right )+\left (-i \sqrt {a}\, s +\sqrt {c}\, b +\sqrt {c}\, n +\sqrt {c}\right ) \KummerM \left (\frac {i \sqrt {a}\, s +\sqrt {c}\, b -\sqrt {c}\, n +\sqrt {c}}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right )}{2 \sqrt {c}\, x a \left (\KummerU \left (\frac {i \sqrt {a}\, s +\sqrt {c}\, b +\sqrt {c}\, n +\sqrt {c}}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right ) c_{1}+\KummerM \left (\frac {i \sqrt {a}\, s +\sqrt {c}\, b +\sqrt {c}\, n +\sqrt {c}}{2 \sqrt {c}\, n}, \frac {b +n +1}{n}, \frac {2 i \sqrt {c}\, \sqrt {a}\, x^{n}}{n}\right )\right )} \]
✓ Solution by Mathematica
Time used: 1.033 (sec). Leaf size: 638
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 ) \text {HypergeometricU}\left (\frac {-\frac {i \sqrt {a} s}{\sqrt {c}}+b+3 n+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 ) \text {HypergeometricU}\left (\frac {-\frac {i \sqrt {a} s}{\sqrt {c}}+b+n+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 \text {HypergeometricU}\left (\frac {-\frac {i \sqrt {a} s}{\sqrt {c}}+b+n+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 ) \text {HypergeometricU}\left (\frac {-\frac {i \sqrt {a} s}{\sqrt {c}}+b+3 n+1}{2 n},\frac {b+2 n+1}{n},-\frac {2 i \sqrt {a} \sqrt {c} x^n}{n}\right )}{n \text {HypergeometricU}\left (\frac {-\frac {i \sqrt {a} s}{\sqrt {c}}+b+n+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*}