53.4.29 problem 32

Internal problem ID [8517]
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 : Wednesday, March 05, 2025 at 06:02:03 AM
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: 0.052 (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: 49.386 (sec). Leaf size: 56

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]
 

\begin{align*} y(x)\to \csc (c_1) \sec (c_1) W\left (\sin (c_1) e^{-\left ((x+c_2) \cos ^2(c_1)\right )-\sin ^2(c_1)}\right )+\tan (c_1) \\ y(x)\to e^{-x-c_2} \\ \end{align*}

✗ 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