67.7.4 problem 4

Internal problem ID [16449]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 8. Review exercises for part of part II. page 143
Problem number : 4
Date solved : Thursday, October 02, 2025 at 01:33:34 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&=\sqrt {x +y} \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 42

ode:=diff(y(x),x) = (x+y(x))^(1/2); 
dsolve(ode,y(x), singsol=all);
 

\[ x -2 \sqrt {x +y}-\ln \left (-1+\sqrt {x +y}\right )+\ln \left (1+\sqrt {x +y}\right )+\ln \left (-1+x +y\right )-c_1 = 0 \]

✓ Mathematica. Time used: 5.384 (sec). Leaf size: 59

ode=D[y[x],x]==Sqrt[x+y[x]]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to W\left (-e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right ){}^2+2 W\left (-e^{-\frac {x}{2}-1-\frac {c_1}{2}}\right )-x+1\\ y(x)&\to 1-x \end{align*}

✓ Sympy. Time used: 0.919 (sec). Leaf size: 27

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

\[ C_{1} + x - 2 \sqrt {x + y{\left (x \right )}} + 2 \log {\left (\sqrt {x + y{\left (x \right )}} + 1 \right )} = 0 \]