29.8.28 problem 28

Internal problem ID [7370]
Book : Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section : Chapter 8, Ordinary differential equations. Section 13. Miscellaneous problems. page 466
Problem number : 28
Date solved : Tuesday, September 30, 2025 at 04:29:57 PM
CAS classification : [[_2nd_order, _missing_x], [_2nd_order, _exact, _nonlinear], [_2nd_order, _reducible, _mu_x_y1], [_2nd_order, _reducible, _mu_xy]]

\begin{align*} y y^{\prime \prime }+{y^{\prime }}^{2}+4&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=3 \\ y^{\prime }\left (1\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.594 (sec). Leaf size: 16

ode:=y(x)*diff(diff(y(x),x),x)+diff(y(x),x)^2+4 = 0; 
ic:=[y(1) = 3, D(y)(1) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \sqrt {-4 x^{2}+8 x +5} \]

✓ Mathematica. Time used: 16.74 (sec). Leaf size: 19

ode=y[x]*D[y[x],{x,2}]+D[y[x],x]^2+4==0; 
ic={y[1]==3,Derivative[1][y][1]==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \sqrt {-4 x^2+8 x+5} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(y(x)*Derivative(y(x), (x, 2)) + Derivative(y(x), x)**2 + 4,0) 
ics = {y(1): 3, Subs(Derivative(y(x), x), x, 1): 0} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE -sqrt(-y(x)*Derivative(y(x), (x, 2)) - 4) + Derivative(y(x), x) cannot be solved by the factorable group method