61.2.14 problem 14

Internal problem ID [12020]
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
Date solved : Monday, January 27, 2025 at 11:51:01 PM
CAS classification : [_rational, _Riccati]

\begin{align*} 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 ) \end{align*}

✓ Solution by Maple

Time used: 0.003 (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)
 

\[ y = \frac {-2 \left (-a \,x^{2}\right )^{b -\frac {1}{2}} {\mathrm e}^{a \,x^{2}} c_{1} a \,x^{2}+\left (a \,x^{2}+b \right ) \left (c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-c_{1} \Gamma \left (b +\frac {1}{2}\right )-1\right )}{x \left (c_{1} \Gamma \left (b +\frac {1}{2}, -a \,x^{2}\right )-c_{1} \Gamma \left (b +\frac {1}{2}\right )-1\right )} \]

✓ Solution by Mathematica

Time used: 0.765 (sec). Leaf size: 128

DSolve[x^2*D[y[x],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*}