87.7.12 problem 15

Internal problem ID [23353]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 57
Problem number : 15
Date solved : Thursday, October 02, 2025 at 09:39:30 PM
CAS classification : [[_2nd_order, _missing_x]]

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

✓ Maple. Time used: 0.012 (sec). Leaf size: 73

ode:=diff(diff(y(x),x),x) = (1+diff(y(x),x)^2)*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: 60.071 (sec). Leaf size: 67

ode=D[y[x],{x,2}]==(1+D[y[x],x]^2)*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: 18.688 (sec). Leaf size: 41

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

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