61.2.47 problem 47

Internal problem ID [12053]
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
Date solved : Monday, January 27, 2025 at 11:54:55 PM
CAS classification : [_rational, _Riccati]

\begin{align*} 2 x^{2} y^{\prime }&=2 y^{2}+y x -2 a^{2} x \end{align*}

✓ Solution by Maple

Time used: 0.006 (sec). Leaf size: 24

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

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

✓ Solution by Mathematica

Time used: 0.447 (sec). Leaf size: 43

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

\[ y(x)\to -\sqrt {-a^2} \sqrt {x} \tan \left (\frac {2 \sqrt {-a^2}}{\sqrt {x}}-c_1\right ) \]