Internal problem ID [9635]
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: 48.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_rational, _Riccati]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime }-2 y^{2}-3 y x +2 x \,a^{2}=0} \end {gather*}
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 100
dsolve(2*x^2*diff(y(x),x)=2*y(x)^2+3*x*y(x)-2*a^2*x,y(x), singsol=all)
\[ y \relax (x ) = \frac {-x \left (c_{1}-2 \sqrt {-\frac {a^{2}}{x}}\right ) \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right )-2 x \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) \left (c_{1} \sqrt {-\frac {a^{2}}{x}}+\frac {1}{2}\right )}{2 \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) c_{1}+2 \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )} \]
✓ Solution by Mathematica
Time used: 0.247 (sec). Leaf size: 66
DSolve[2*x^2*y'[x]==2*y[x]^2+3*x*y[x]-2*a^2*x,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {4 a^2 c_1 \sqrt {x}}{e^{\frac {4 a}{\sqrt {x}}}-2 a c_1}+a \sqrt {x}-\frac {x}{2} \\ y(x)\to a \left (-\sqrt {x}\right )-\frac {x}{2} \\ \end{align*}