12.1.27 problem Problem 14.31

Internal problem ID [3483]
Book : Mathematical methods for physics and engineering, Riley, Hobson, Bence, second edition, 2002
Section : Chapter 14, First order ordinary differential equations. 14.4 Exercises, page 490
Problem number : Problem 14.31
Date solved : Tuesday, September 30, 2025 at 06:40:36 AM
CAS classification : [[_2nd_order, _missing_x], _Liouville, [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y^{\prime \prime }+{y^{\prime }}^{2}+y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.056 (sec). Leaf size: 18

ode:=diff(diff(y(x),x),x)+diff(y(x),x)^2+diff(y(x),x) = 0; 
ic:=[y(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \ln \left (c_2 \,{\mathrm e}^{x}-c_2 +1\right )-x \]

✓ Mathematica. Time used: 0.224 (sec). Leaf size: 54

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

\begin{align*} y(x)&\to \log \left (-e^x\right )-\log \left (e^x\right )-i \pi \\ y(x)&\to -\log \left (e^x\right )+\log \left (-e^x+e^{c_1}\right )-\log \left (-1+e^{c_1}\right ) \end{align*}

✓ Sympy. Time used: 0.598 (sec). Leaf size: 15

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

\[ y{\left (x \right )} = - x - \log {\left (C_{2} + 1 \right )} + \log {\left (C_{2} + e^{x} \right )} \]