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 : Monday, January 27, 2025 at 03:12:24 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*}

✓ Solution by Maple

Time used: 0.961 (sec). Leaf size: 25

dsolve([diff(y(x),x$2)=-1/(2*diff(y(x),x)^2),y(0) = 1, D(y)(0) = -1],y(x), singsol=all)
 

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

✓ Solution by Mathematica

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

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

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