15.18.9 problem 9

Internal problem ID [3252]
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 : 9
Date solved : Sunday, March 30, 2025 at 01:25:08 AM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 73

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

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

✓ Mathematica. Time used: 60.135 (sec). Leaf size: 67

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

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

✓ Sympy. Time used: 26.449 (sec). Leaf size: 41

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)
 

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