55.2.18 problem 18

Internal problem ID [13244]
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 : 18
Date solved : Wednesday, October 01, 2025 at 04:05:19 AM
CAS classification : [_rational, [_Riccati, _special]]

\begin{align*} x^{4} y^{\prime }&=-x^{4} y^{2}-a^{2} \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 24

ode:=x^4*diff(y(x),x) = -x^4*y(x)^2-a^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {-a \tan \left (\frac {a \left (c_1 x -1\right )}{x}\right )+x}{x^{2}} \]

✓ Mathematica. Time used: 0.345 (sec). Leaf size: 94

ode=x^4*D[y[x],x]==-x^4*y[x]^2-a^2; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {2 a^2 c_1 e^{\frac {2 i a}{x}}+i a \left (e^2+2 c_1 x e^{\frac {2 i a}{x}}\right )+e^2 x}{x^2 \left (e^2+2 i a c_1 e^{\frac {2 i a}{x}}\right )}\\ y(x)&\to \frac {x-i a}{x^2} \end{align*}

✓ Sympy. Time used: 2.835 (sec). Leaf size: 58

from sympy import * 
x = symbols("x") 
a = symbols("a") 
y = Function("y") 
ode = Eq(a**2 + x**4*y(x)**2 + x**4*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {i a e^{\frac {2 i a \left (C_{1} x - 1\right )}{x}} + i a + x e^{\frac {2 i a \left (C_{1} x - 1\right )}{x}} - x}{x^{2} \left (e^{\frac {2 i a \left (C_{1} x - 1\right )}{x}} - 1\right )} \]