71.8.44 problem 14 (d)

Internal problem ID [14399]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.4.4, page 115
Problem number : 14 (d)
Date solved : Saturday, February 22, 2025 at 03:46:45 PM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y^{\prime }&=-\frac {x}{2}+\frac {\sqrt {x^{2}+4 y}}{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-{\frac {1}{5}} \end{align*}

✓ Maple. Time used: 1.751 (sec). Leaf size: 69

ode:=diff(y(x),x) = -1/2*x+1/2*(x^2+4*y(x))^(1/2); 
ic:=y(1) = -1/5; 
dsolve([ode,ic],y(x), singsol=all);
 

\begin{align*} y &= \frac {\left (x -1\right ) \sqrt {5}}{10}-\frac {x}{2}+\frac {3}{10} \\ y &= \frac {\left (-5+\sqrt {5}\right ) \left (-5+\sqrt {5}+10 x \right )}{100} \\ y &= -\frac {\left (2^{{1}/{3}} x -\frac {\left (50+20 \sqrt {5}\right )^{{1}/{3}}}{5}\right ) \left (50+20 \sqrt {5}\right )^{{1}/{3}} 2^{{1}/{3}}}{10} \\ \end{align*}

✓ Mathematica. Time used: 0.478 (sec). Leaf size: 51

ode=D[y[x],x]==(-x+Sqrt[x^2+4*y[x]])/2; 
ic={y[1]==-2/10}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{100} \left (5+\sqrt {5}\right ) \left (-10 x+\sqrt {5}+5\right ) \\ y(x)\to \frac {1}{100} \left (\sqrt {5}-5\right ) \left (10 x+\sqrt {5}-5\right ) \\ \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x/2 - sqrt(x**2 + 4*y(x))/2 + Derivative(y(x), x),0) 
ics = {y(1): -1/5} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : Initial conditions produced too many solutions for constants