Internal problem ID [9601]
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: 14.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {x^{2} y^{\prime }-x^{2} y^{2}+a^{2} x^{4}-a \left (1-2 b \right ) x^{2}+b \left (b +1\right )=0} \end {gather*}
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 121
dsolve(x^2*diff(y(x),x)=x^2*y(x)^2-a^2*x^4+a*(1-2*b)*x^2-b*(b+1),y(x), singsol=all)
\[ y \relax (x ) = \frac {2 \left (-a \,x^{2}\right )^{b -\frac {1}{2}} c_{1} a x \,{\mathrm e}^{a \,x^{2}}}{c_{1} \Gamma \left (b +\frac {1}{2}\right )-c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )+1}+\frac {\left (-a \,x^{2} c_{1}-c_{1} b \right ) \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )+\left (a \,x^{2} c_{1}+c_{1} b \right ) \Gamma \left (b +\frac {1}{2}\right )+a \,x^{2}+b}{x \left (c_{1} \Gamma \left (b +\frac {1}{2}\right )-c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )+1\right )} \]
✓ Solution by Mathematica
Time used: 0.603 (sec). Leaf size: 106
DSolve[x^2*y'[x]==x^2*y[x]^2-a^2*x^4+a*(1-2*b)*x^2-b*(b+1),y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {x^{2 b+1} \left (a x^2+b\right ) E_{\frac {1}{2}-b}\left (-a x^2\right )-2 c_1 \left (a x^2+b\right )+2 e^{a x^2} x^{2 b+1}}{x^{2 b+2} E_{\frac {1}{2}-b}\left (-a x^2\right )-2 c_1 x} \\ y(x)\to a x+\frac {b}{x} \\ \end{align*}