9.4.14 problem 15

Internal problem ID [2927]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 8, page 34
Problem number : 15
Date solved : Tuesday, September 30, 2025 at 06:08:37 AM
CAS classification : [[_homogeneous, `class G`], _exact, _rational, [_Abel, `2nd type`, `class B`]]

\begin{align*} \frac {y \left (2+x^{3} y\right )}{x^{3}}&=\frac {\left (1-2 x^{3} y\right ) y^{\prime }}{x^{2}} \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 57

ode:=y(x)*(2+x^3*y(x))/x^3 = (1-2*x^3*y(x))/x^2*diff(y(x),x); 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {c_1^{3}-\sqrt {c_1^{6}+4 c_1 \,x^{5}}}{2 x^{3} c_1^{3}} \\ y &= \frac {c_1^{3}+\sqrt {c_1^{6}+4 c_1 \,x^{5}}}{2 x^{3} c_1^{3}} \\ \end{align*}

✓ Mathematica. Time used: 0.675 (sec). Leaf size: 80

ode=y[x]*(2+x^3*y[x])/x^3==(1-2*x^3*y[x])/x^2*D[y[x],x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1-\sqrt {\frac {1}{x^3}} x^2 \sqrt {\frac {1}{x}+4 c_1 x^4}}{2 x^3}\\ y(x)&\to \frac {1+\sqrt {\frac {1}{x^3}} x^2 \sqrt {\frac {1}{x}+4 c_1 x^4}}{2 x^3} \end{align*}

✓ Sympy. Time used: 0.632 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x**3*y(x) - 1)*Derivative(y(x), x)/x**2 + (x**3*y(x) + 2)*y(x)/x**3,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {1 - \sqrt {C_{1} x^{5} + 1}}{2 x^{3}}, \ y{\left (x \right )} = \frac {\sqrt {C_{1} x^{5} + 1} + 1}{2 x^{3}}\right ] \]