55.2.48 problem 48

Internal problem ID [13274]
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 : Wednesday, October 01, 2025 at 05:02:26 AM
CAS classification : [_rational, _Riccati]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 102

ode:=2*x^2*diff(y(x),x) = 2*y(x)^2+3*x*y(x)-2*a^2*x; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {\left (-2 x c_1 \sqrt {-\frac {a^{2}}{x}}-x \right ) \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )-x \left (c_1 -2 \sqrt {-\frac {a^{2}}{x}}\right ) \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right )}{2 \cos \left (2 \sqrt {-\frac {a^{2}}{x}}\right ) c_1 +2 \sin \left (2 \sqrt {-\frac {a^{2}}{x}}\right )} \]

✓ Mathematica. Time used: 0.194 (sec). Leaf size: 94

ode=2*x^2*D[y[x],x]==2*y[x]^2+3*x*y[x]-2*a^2*x; 
ic={}; 
DSolve[{ode,ic},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*}

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(2*a**2*x + 2*x**2*Derivative(y(x), x) - 3*x*y(x) - 2*y(x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

RecursionError : maximum recursion depth exceeded