85.27.12 problem 12

Internal problem ID [22604]
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 : 12
Date solved : Thursday, October 02, 2025 at 08:54:56 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

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

✓ Maple. Time used: 0.024 (sec). Leaf size: 28

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

\begin{align*} y &= 0 \\ y &= {\mathrm e}^{\operatorname {RootOf}\left (c_2 \,{\mathrm e}^{c_1}+x \,{\mathrm e}^{c_1}+\operatorname {Ei}_{1}\left (-\textit {\_Z} -c_1 \right )\right )} \\ \end{align*}

✓ Mathematica. Time used: 0.145 (sec). Leaf size: 77

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

\begin{align*} y(x)&\to \text {InverseFunction}\left [e^{-c_1} \operatorname {ExpIntegralEi}(c_1+\log (\text {$\#$1}))\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [e^{-(-c_1)} \operatorname {ExpIntegralEi}(\log (\text {$\#$1})-c_1)\&\right ][x+c_2]\\ y(x)&\to \text {InverseFunction}\left [e^{-c_1} \operatorname {ExpIntegralEi}(c_1+\log (\text {$\#$1}))\&\right ][x+c_2] \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x) cannot be s