53.4.37 problem 40

Internal problem ID [8525]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 101. Independent variable missing. EXERCISES Page 324
Problem number : 40
Date solved : Sunday, March 30, 2025 at 01:13:21 PM
CAS classification : [[_2nd_order, _missing_y]]

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

✓ Maple. Time used: 0.099 (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: 0.004 (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]
 

\[ y(x)\to \frac {c_1 x^2}{2}-c_1{}^2 x+c_2 \]

✓ Sympy. Time used: 2.056 (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)
 

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