Internal problem ID [9589]
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: 6.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
\[ \boxed {y^{\prime }-y^{2} a -b \,x^{2 n}-c \,x^{n -1}=0} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 499
dsolve(diff(y(x),x)=a*y(x)^2+b*x^(2*n)+c*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (-2 b^{\frac {3}{2}} c_{1} n -2 b^{\frac {3}{2}} c_{1} \right ) \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right )}{2 b^{\frac {3}{2}} \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right )\right ) a x}-\frac {\left (2 i \sqrt {a}\, x^{1+n} c_{1} b^{2}+i \sqrt {a}\, c_{1} b c -b^{\frac {3}{2}} c_{1} n \right ) \operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right )+\left (-i \sqrt {a}\, b c +b^{\frac {3}{2}} n +2 b^{\frac {3}{2}}\right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c -2 \sqrt {b}\, n -2 \sqrt {b}}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right )+\left (2 i \sqrt {a}\, x^{1+n} b^{2}+i \sqrt {a}\, b c -b^{\frac {3}{2}} n \right ) \operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right )}{2 b^{\frac {3}{2}} \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{2 \sqrt {b}\, \left (1+n \right )}, \frac {1}{2+2 n}, \frac {2 i \sqrt {b}\, \sqrt {a}\, x^{1+n}}{1+n}\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 0.981 (sec). Leaf size: 982
DSolve[y'[x]==a*y[x]^2+b*x^(2*n)+c*x^(n-1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to -\frac {x^n \left (\sqrt {b} c_1 (n+1) \sqrt {-(n+1)^2} \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \left (\sqrt {a} c (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {3 n+2}{n+1}\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+\sqrt {b} (n+1) \sqrt {-(n+1)^2} \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+2 L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {3 n+2}{2 n+2}}^{\frac {n}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )\right )}{\sqrt {a} (n+1)^2 \left (L_{-\frac {\sqrt {a} c}{2 \sqrt {b} \sqrt {-(n+1)^2}}-\frac {n}{2 (n+1)}}^{-\frac {1}{n+1}}\left (\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )+c_1 \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )\right )} \\ y(x)\to \frac {x^n \left (-\frac {\left (\sqrt {a} c (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}-\sqrt {b} \sqrt {-(n+1)^2} (n+1)\right )}{\sqrt {a} (n+1)^2} \\ y(x)\to \frac {x^n \left (-\frac {\left (\sqrt {a} c (n+1)+\sqrt {b} \sqrt {-(n+1)^2} n\right ) \operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}+2\right ),\frac {n}{n+1}+1,\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}{\operatorname {HypergeometricU}\left (\frac {1}{2} \left (\frac {\sqrt {a} c}{\sqrt {b} \sqrt {-(n+1)^2}}+\frac {n}{n+1}\right ),\frac {n}{n+1},\frac {2 \sqrt {a} \sqrt {b} x^{n+1}}{\sqrt {-(n+1)^2}}\right )}-\sqrt {b} \sqrt {-(n+1)^2} (n+1)\right )}{\sqrt {a} (n+1)^2} \\ \end{align*}