Internal problem ID [9592]
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: 5.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_Riccati]
Solve \begin {gather*} \boxed {y^{\prime }-y^{2}-a n \,x^{n -1}+a^{2} x^{2 n}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 555
dsolve(diff(y(x),x)=y(x)^2+a*n*x^(n-1)-a^2*x^(2*n),y(x), singsol=all)
\[ y \relax (x ) = \frac {2 a \,x^{n +1} {\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}}+\left (2 x^{-\frac {3 n}{2}-1} c_{1} n^{3}+11 x^{-\frac {3 n}{2}-1} c_{1} n^{2}+20 x^{-\frac {3 n}{2}-1} c_{1} n +12 x^{-\frac {3 n}{2}-1} c_{1}\right ) \WhittakerM \left (\frac {3 n +4}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n +1}}{n +1}\right )+\left (-3 x^{-\frac {n}{2}} c_{1} a \,n^{2}-10 x^{-\frac {n}{2}} c_{1} a n -x^{-\frac {3 n}{2}-1} c_{1} n^{3}-8 x^{-\frac {n}{2}} c_{1} a -5 x^{-\frac {3 n}{2}-1} c_{1} n^{2}-8 x^{-\frac {3 n}{2}-1} c_{1} n -4 x^{-\frac {3 n}{2}-1} c_{1}\right ) \WhittakerM \left (\frac {2+n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n +1}}{n +1}\right )+\left (-x^{-\frac {n}{2}} c_{1} a \,n^{2}+2 x^{1+\frac {n}{2}} c_{1} a^{2} n -3 x^{-\frac {n}{2}} c_{1} a n +2 x^{1+\frac {n}{2}} c_{1} a^{2}-x^{-\frac {3 n}{2}-1} c_{1} n^{3}-2 x^{-\frac {n}{2}} c_{1} a -4 x^{-\frac {3 n}{2}-1} c_{1} n^{2}-5 x^{-\frac {3 n}{2}-1} c_{1} n -2 x^{-\frac {3 n}{2}-1} c_{1}\right ) \WhittakerM \left (-\frac {n}{2 \left (n +1\right )}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n +1}}{n +1}\right )}{\left (2 \,{\mathrm e}^{-\frac {a \,x^{n +1}}{n +1}}+\left (-x^{-\frac {3 n}{2}-1} c_{1} n^{2}-4 x^{-\frac {3 n}{2}-1} c_{1} n -4 x^{-\frac {3 n}{2}-1} c_{1}\right ) \WhittakerM \left (\frac {2+n}{2 n +2}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n +1}}{n +1}\right )+\left (-x^{-\frac {3 n}{2}-1} c_{1} n^{2}+2 x^{-\frac {n}{2}} c_{1} a n -3 x^{-\frac {3 n}{2}-1} c_{1} n +2 x^{-\frac {n}{2}} c_{1} a -2 x^{-\frac {3 n}{2}-1} c_{1}\right ) \WhittakerM \left (-\frac {n}{2 \left (n +1\right )}, \frac {2 n +3}{2 n +2}, -\frac {2 a \,x^{n +1}}{n +1}\right )\right ) x} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[x]==y[x]^2+a*n*x^(n-1)-a^2*x^(2*n),y[x],x,IncludeSingularSolutions -> True]
Not solved