87.7.3 problem 3

Internal problem ID [23344]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 57
Problem number : 3
Date solved : Thursday, October 02, 2025 at 09:39:21 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _reducible, _mu_x_y1]]

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

With initial conditions

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

✓ Maple. Time used: 0.115 (sec). Leaf size: 14

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

\[ y = \frac {1}{2} x^{2}-x +1 \]

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 17

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

\begin{align*} y(x)&\to \frac {1}{2} \left (x^2-2 x+2\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(Derivative(y(x), x)**2 + 1)/(2*y(x)) + 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)
 

NotImplementedError : The given ODE -sqrt(2*y(x)*Derivative(y(x), (x, 2)) - 1) + Derivative(y(x), x)