23.4.50 problem 50

Internal problem ID [6352]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 4. THE NONLINEAR EQUATION OF SECOND ORDER, page 380
Problem number : 50
Date solved : Tuesday, September 30, 2025 at 02:52:21 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.076 (sec). Leaf size: 35

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

\begin{align*} y &= \arctan \left (\sqrt {{\mathrm e}^{2 x} c_1 -1}\right )+c_2 \\ y &= -\arctan \left (\sqrt {{\mathrm e}^{2 x} c_1 -1}\right )+c_2 \\ \end{align*}

✓ Mathematica. Time used: 60.07 (sec). Leaf size: 69

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

\begin{align*} y(x)&\to c_2-\arctan \left (e^{-c_1} \sqrt {e^{2 x}-e^{2 c_1}}\right )\\ y(x)&\to \arctan \left (e^{-c_1} \sqrt {e^{2 x}-e^{2 c_1}}\right )+c_2 \end{align*}

✓ Sympy. Time used: 25.029 (sec). Leaf size: 112

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

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