2.47 problem 47

Internal problem ID [9634]

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: 47.
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}-y x +2 a^{2} x=0} \end {gather*}

Solution by Maple

Time used: 0.005 (sec). Leaf size: 27

dsolve(2*x^2*diff(y(x),x)=2*y(x)^2+x*y(x)-2*a^2*x,y(x), singsol=all)
 

\[ y \relax (x ) = -i \tan \left (\frac {2 i a -c_{1} \sqrt {x}}{\sqrt {x}}\right ) \sqrt {x}\, a \]

Solution by Mathematica

Time used: 0.65 (sec). Leaf size: 43

DSolve[2*x^2*y'[x]==2*y[x]^2+x*y[x]-2*a^2*x,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-a^2} \sqrt {x} \tan \left (\frac {2 \sqrt {-a^2}}{\sqrt {x}}-c_1\right ) \\ \end{align*}