86.8.4 problem 4

Internal problem ID [23164]
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 : 4
Date solved : Thursday, October 02, 2025 at 09:23:39 PM
CAS classification : [[_2nd_order, _missing_y], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_y_y1], [_2nd_order, _reducible, _mu_poly_yn]]

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

With initial conditions

\begin{align*} y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.089 (sec). Leaf size: 69

ode:=diff(diff(y(x),x),x)*diff(y(x),x)^2-x^2 = 0; 
ic:=[D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\begin{align*} y &= \int _{0}^{x}\left (\textit {\_z1}^{3}\right )^{{1}/{3}}d \textit {\_z1} +c_2 \\ y &= \int _{0}^{x}-\frac {\left (\textit {\_z1}^{3}\right )^{{1}/{3}} \left (1+i \sqrt {3}\right )}{2}d \textit {\_z1} +c_2 \\ y &= \int _{0}^{x}\frac {\left (\textit {\_z1}^{3}\right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{2}d \textit {\_z1} +c_2 \\ \end{align*}

✓ Mathematica. Time used: 1.738 (sec). Leaf size: 179

ode=D[y[x],{x,2}]*D[y[x],x]^2-x^2==0; 
ic={Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\operatorname {Gamma}\left (-\frac {1}{3}\right ) \left (\left (1-i \sqrt {3}\right ) x \sqrt [3]{x^3}-4 c_2\right )}{18 \operatorname {Gamma}\left (\frac {5}{3}\right )}\\ y(x)&\to \frac {\operatorname {Gamma}\left (-\frac {1}{3}\right ) \left (\left (1+i \sqrt {3}\right ) x \sqrt [3]{x^3}-4 c_2\right )}{18 \operatorname {Gamma}\left (\frac {5}{3}\right )}\\ y(x)&\to \frac {\operatorname {Gamma}\left (\frac {2}{3}\right ) \left (\sqrt [3]{x^3} x+2 c_2\right )}{3 \operatorname {Gamma}\left (\frac {5}{3}\right )}\\ y(x)&\to 0 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},\text {ComplexInfinity}\right )+c_2\\ y(x)&\to 0 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},\text {ComplexInfinity}\right )+c_2\\ y(x)&\to 0 \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {1}{3},\frac {4}{3},\text {ComplexInfinity}\right )+c_2 \end{align*}

✗ Sympy

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 = {Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out