Internal problem ID [10336]
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]
\[ \boxed {y^{\prime }-a y^{2}=b \,x^{2 n}+c \,x^{n -1}} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 357
dsolve(diff(y(x),x)=a*y(x)^2+b*x^(2*n)+c*x^(n-1),y(x), singsol=all)
\[ y \left (x \right ) = -\frac {\left (\left (\frac {n}{2}+1\right ) \sqrt {b}-\frac {i c \sqrt {a}}{2}\right ) \operatorname {WhittakerM}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i c \sqrt {a}}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )-\sqrt {b}\, c_{1} \left (n +1\right ) \operatorname {WhittakerW}\left (-\frac {\left (-2 n -2\right ) \sqrt {b}+i c \sqrt {a}}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )+\left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )\right ) \left (-\frac {\sqrt {b}\, n}{2}+i \sqrt {a}\, \left (x \,x^{n} b +\frac {c}{2}\right )\right )}{\sqrt {b}\, \left (\operatorname {WhittakerW}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right ) c_{1} +\operatorname {WhittakerM}\left (-\frac {i \sqrt {a}\, c}{\sqrt {b}\, \left (2 n +2\right )}, \frac {1}{2 n +2}, \frac {2 i \sqrt {a}\, \sqrt {b}\, x \,x^{n}}{n +1}\right )\right ) a x} \]
✓ Solution by Mathematica
Time used: 1.818 (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*}