54.7.64 problem 1672 (book 6.81)

Internal problem ID [12913]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 6, non-linear second order
Problem number : 1672 (book 6.81)
Date solved : Wednesday, October 01, 2025 at 02:46:10 AM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

\begin{align*} 2 x y^{\prime \prime }+{y^{\prime }}^{3}+y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 41

ode:=2*x*diff(diff(y(x),x),x)+diff(y(x),x)^3+diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {c_2 c_1 +2 \sqrt {c_1 x -1}}{c_1} \\ y &= \frac {c_2 c_1 -2 \sqrt {c_1 x -1}}{c_1} \\ \end{align*}

✓ Mathematica. Time used: 0.584 (sec). Leaf size: 48

ode=D[y[x],x] + D[y[x],x]^3 + 2*x*D[y[x],{x,2}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _1^x\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1] \left (K[1]^2+1\right )}dK[1]\&\right ]\left [c_1-\frac {1}{2} \log (K[2])\right ]dK[2]+c_2 \end{align*}

✓ Sympy. Time used: 6.625 (sec). Leaf size: 61

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

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