89.33.37 problem 40

Internal problem ID [25021]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 17. Special Equations of order Two. Exercises at page 251
Problem number : 40
Date solved : Thursday, October 02, 2025 at 11:47:22 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.152 (sec). Leaf size: 28

ode:=diff(diff(y(x),x),x)^2-x*diff(diff(y(x),x),x)+diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {x^{3}}{12}+c_{1} \\ y &= \frac {1}{2} c_{1} x^{2}-c_{1}^{2} x +c_{2} \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 24

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

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

✓ Sympy. Time used: 1.263 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = C_{1} - \frac {C_{2}^{2} x}{4} - \frac {C_{2} x^{2}}{4} \]