85.27.14 problem 14

Internal problem ID [22606]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter two. First order and simple higher order ordinary differential equations. A Exercises at page 60
Problem number : 14
Date solved : Thursday, October 02, 2025 at 08:54:59 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.020 (sec). Leaf size: 32

ode:=diff(diff(y(x),x),x) = diff(y(x),x)*(1+y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (\sqrt {2}\, c_1 -2 \tan \left (\frac {\left (x +c_2 \right ) \sqrt {2}}{2 c_1}\right )\right ) \sqrt {2}}{2 c_1} \]

✓ Mathematica. Time used: 60.023 (sec). Leaf size: 37

ode=D[y[x],{x,2}]==D[y[x],{x,1}]*(1+y[x]); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -1+\sqrt {-1+2 c_1} \tan \left (\frac {1}{2} \sqrt {-1+2 c_1} (x+c_2)\right ) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - Derivative(y(x), (x, 2))/(y(x) + 1) cannot