61.2.48 problem 48

Internal problem ID [12054]
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
Date solved : Monday, January 27, 2025 at 11:54:56 PM
CAS classification : [_rational, _Riccati]

\begin{align*} 2 x^{2} y^{\prime }&=2 y^{2}+3 y x -2 a^{2} x \end{align*}

✓ Solution by Maple

Time used: 0.027 (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)
 

\[ y = \frac {\left (-2 c_{1} \sqrt {-\frac {a^{2}}{x}}\, x -x \right ) \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )-x \left (c_{1} -2 \sqrt {-\frac {a^{2}}{x}}\right ) \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right )}{2 \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) c_{1} +2 \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )} \]

✓ Solution by Mathematica

Time used: 0.287 (sec). Leaf size: 94

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