47.4.29 problem 32

Internal problem ID [9803]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 32
Date solved : Tuesday, September 30, 2025 at 06:42:02 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

\begin{align*} \left (1+y^{2}\right ) y^{\prime \prime }+{y^{\prime }}^{3}+y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.208 (sec). Leaf size: 113

ode:=(1+y(x)^2)*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 &= -i \\ y &= i \\ y &= c_1 \\ y &= \frac {i c_1 -{\mathrm e}^{\frac {-4 \operatorname {LambertW}\left (-\frac {i {\mathrm e}^{\frac {\left (-c_2 -x +1\right ) c_1^{2}+\left (-2 c_2 -2 x -2\right ) c_1 -x -c_2 +1}{4 c_1}} \left (c_1 -1\right )}{4 c_1}\right ) c_1 +\left (-c_2 -x +1\right ) c_1^{2}+\left (-2 c_2 -2 x -2\right ) c_1 -x -c_2 +1}{4 c_1}}-i}{c_1 +1} \\ \end{align*}

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 57

ode=(1+y[x]^2)*D[y[x],{x,2}]+(D[y[x],x])^3+D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\frac {1}{\text {InverseFunction}\left [\int _1^{\text {$\#$1}}\frac {1}{K[1]^2+1}dK[1]\&\right ]\left [c_1+\int _1^{K[3]}-\frac {1}{K[2]^2+1}dK[2]\right ]}dK[3]=x+c_2,y(x)\right ] \]

✗ Sympy

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

NotImplementedError : The given ODE (sqrt((27*y(x)**2*Derivative(y(x), (x, 2)) + 27*Derivative(y(x), (x, 2)))**2 + 108)/2 + 27*y(x)**2*Derivative(y(x), (x, 2))/2 + 27*Derivative(y(x), (x, 2))/2)**(1/3)/3 + Derivative(y(x), x) - 1/(sqrt((27*y(x)**2*Derivative(y(x), (x, 2)) + 27*Derivative(y(x), (x, 2)))**2 + 108)/2 + 27*y(x)**2*Derivative(y(x), (x, 2))/2 + 27*Derivative(y(x), (x, 2))/2)**(1/3) cannot be solved by the factorable group method