49.23.8 problem 3

Internal problem ID [7766]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 6. Existence and uniqueness of solutions to systems and nth order equations. Page 238
Problem number : 3
Date solved : Sunday, March 30, 2025 at 12:23:02 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_poly_yn]]

\begin{align*} y^{\prime \prime }&=-\frac {1}{2 {y^{\prime }}^{2}} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=1\\ y^{\prime }\left (0\right )&=-1 \end{align*}

✓ Maple. Time used: 0.268 (sec). Leaf size: 25

ode:=diff(diff(y(x),x),x) = -1/2/diff(y(x),x)^2; 
ic:=y(0) = 1, D(y)(0) = -1; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {3 \left (\frac {2}{3}+x \right ) \left (-8-12 x \right )^{{1}/{3}} \left (i \sqrt {3}-1\right )}{16}+\frac {3}{2} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 27

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

\[ y(x)\to \frac {1}{8} \left (12-(-2)^{2/3} (-3 x-2)^{4/3}\right ) \]

✓ Sympy. Time used: 7.560 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(Derivative(y(x), (x, 2)) + 1/(2*Derivative(y(x), x)**2),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - \frac {2^{\frac {2}{3}} i \left (\sqrt {3} + i\right ) \left (- 3 x - 2\right )^{\frac {4}{3}}}{16} + 1 - \frac {\left (-1\right )^{\frac {5}{6}} \sqrt {3}}{4} + \frac {\sqrt [3]{-1}}{4} \]