15.18.27 problem 27

Internal problem ID [3270]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 35, page 157
Problem number : 27
Date solved : Sunday, March 30, 2025 at 01:25:53 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 37

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

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

✓ Mathematica. Time used: 60.114 (sec). Leaf size: 47

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

\begin{align*} y(x)\to c_2-\text {arctanh}\left (\sqrt {1+e^{2 (x+c_1)}}\right ) \\ y(x)\to \text {arctanh}\left (\sqrt {1+e^{2 (x+c_1)}}\right )+c_2 \\ \end{align*}

✓ Sympy. Time used: 42.356 (sec). Leaf size: 102

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)
 

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