Internal problem ID [10388]
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]
\[ \boxed {2 y^{\prime } x^{2}-2 y^{2}-3 y x=-2 a^{2} x} \]
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 102
dsolve(2*x^2*diff(y(x),x)=2*y(x)^2+3*x*y(x)-2*a^2*x,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\left (-2 c_{1} x \sqrt {-\frac {a^{2}}{x}}-x \right ) \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )-x \left (c_{1} -2 \sqrt {-\frac {a^{2}}{x}}\right ) \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right )}{2 \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) c_{1} +2 \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )} \]
✓ Solution by Mathematica
Time used: 0.457 (sec). Leaf size: 94
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}+2 a \sqrt {x} e^{\frac {4 a}{\sqrt {x}}}-x e^{\frac {4 a}{\sqrt {x}}}+2 a c_1 x}{2 e^{\frac {4 a}{\sqrt {x}}}-4 a c_1} y(x)\to a \left (-\sqrt {x}\right )-\frac {x}{2} \end{align*}