2.5.16 problem 16

Internal problem ID [744]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 1.6, Substitution methods and exact equations. Page 74
Problem number : 16
Date solved : Tuesday, September 30, 2025 at 04:08:19 AM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

\begin{align*} y^{\prime }&=\sqrt {1+x +y} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 44

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

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

✓ Mathematica. Time used: 5.451 (sec). Leaf size: 56

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

\begin{align*} y(x)&\to W\left (-e^{\frac {1}{2} (-x-3-c_1)}\right ){}^2+2 W\left (-e^{\frac {1}{2} (-x-3-c_1)}\right )-x\\ y(x)&\to -x \end{align*}

✓ Sympy. Time used: 0.532 (sec). Leaf size: 31

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

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