54.1.178 problem 181

Internal problem ID [11492]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 181
Date solved : Tuesday, September 30, 2025 at 08:42:51 PM
CAS classification : [_rational, [_Riccati, _special]]

\begin{align*} x^{4} \left (y^{\prime }+y^{2}\right )+a&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 28

ode:=x^4*(diff(y(x),x)+y(x)^2)+a = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.255 (sec). Leaf size: 116

ode=x^4*(D[y[x],x]+y[x]^2) + a==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 10.339 (sec). Leaf size: 66

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

\[ C_{1} - \frac {\sqrt {- \frac {1}{a}} \log {\left (- a \sqrt {- \frac {1}{a}} + x \left (x y{\left (x \right )} - 1\right ) \right )}}{2} + \frac {\sqrt {- \frac {1}{a}} \log {\left (a \sqrt {- \frac {1}{a}} + x \left (x y{\left (x \right )} - 1\right ) \right )}}{2} - \frac {1}{x} = 0 \]