Internal problem ID [10344]
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]
\[ \boxed {x^{2} y^{\prime }-x^{2} y^{2}=-a^{2} x^{4}+a \left (1-2 b \right ) x^{2}-b \left (b +1\right )} \]
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 84
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)
\[ y \left (x \right ) = \frac {-2 \left (-a \,x^{2}\right )^{b -\frac {1}{2}} {\mathrm e}^{a \,x^{2}} c_{1} a \,x^{2}+\left (-c_{1} \Gamma \left (b +\frac {1}{2}\right )+c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-1\right ) \left (a \,x^{2}+b \right )}{x \left (-c_{1} \Gamma \left (b +\frac {1}{2}\right )+c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-1\right )} \]
✓ Solution by Mathematica
Time used: 1.225 (sec). Leaf size: 128
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 ) \Gamma \left (b+\frac {1}{2},-a x^2\right )-2 \left (-a x^2\right )^{b+\frac {1}{2}} \left (-e^{a x^2} x^{2 b+1}+c_1 \left (a x^2+b\right )\right )}{x^{2 b+2} \Gamma \left (b+\frac {1}{2},-a x^2\right )-2 c_1 x \left (-a x^2\right )^{b+\frac {1}{2}}} \\ y(x)\to a x+\frac {b}{x} \\ \end{align*}