86.8.5 problem 5

Internal problem ID [23165]
Book : An introduction to Differential Equations. By Howard Frederick Cleaves. 1969. Oliver and Boyd publisher. ISBN 0050015044
Section : Chapter 5. Linear equations of the second order with constant coefficients. Exercise 5e at page 91
Problem number : 5
Date solved : Thursday, October 02, 2025 at 09:23:40 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _reducible, _mu_y_y1]]

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

✓ Maple. Time used: 0.010 (sec). Leaf size: 23

ode:=x^2*diff(diff(y(x),x),x) = diff(y(x),x)^2; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {x}{c_1}-\frac {\ln \left (c_1 x -1\right )}{c_1^{2}}+c_2 \]

✓ Mathematica. Time used: 0.33 (sec). Leaf size: 49

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

\begin{align*} y(x)&\to -\frac {x}{c_1}-\frac {\log (1-c_1 x)}{c_1{}^2}+c_2\\ y(x)&\to c_2\\ y(x)&\to \frac {x^2}{2}+c_2 \end{align*}

✓ Sympy. Time used: 0.406 (sec). Leaf size: 17

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

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