65.8.45 problem 14 (e)

Internal problem ID [15766]
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 (e)
Date solved : Thursday, October 02, 2025 at 10:27:39 AM
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}{4}} \\ \end{align*}

✓ Maple. Time used: 0.648 (sec). Leaf size: 17

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

\begin{align*} y &= -\frac {x^{2}}{4} \\ y &= \frac {1}{4}-\frac {x}{2} \\ \end{align*}

✓ Mathematica. Time used: 0.394 (sec). Leaf size: 14

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

\begin{align*} y(x)&\to \frac {1}{4} (1-2 x) \end{align*}

✓ Sympy. Time used: 0.494 (sec). Leaf size: 8

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): -1/4} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {1}{4} - \frac {x}{2} \]