60.1.324 problem 330

Internal problem ID [10338]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 1, linear first order
Problem number : 330
Date solved : Sunday, March 30, 2025 at 04:12:30 PM
CAS classification : [[_homogeneous, `class C`], _exact, _dAlembert]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 22

ode:=(f(x+y(x))+1)*diff(y(x),x)+f(x+y(x)) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = -x +\operatorname {RootOf}\left (-x +\int _{}^{\textit {\_Z}}\left (f \left (\textit {\_a} \right )+1\right )d \textit {\_a} +c_1 \right ) \]

✓ Mathematica. Time used: 0.095 (sec). Leaf size: 52

ode=f[x + y[x]] + (1 + f[x + y[x]])*D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{y(x)}\left (f(x+K[2])-\int _1^xf''(K[1]+K[2])dK[1]+1\right )dK[2]+\int _1^xf(K[1]+y(x))dK[1]=c_1,y(x)\right ] \]

✓ Sympy. Time used: 0.785 (sec). Leaf size: 17

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

\[ y{\left (x \right )} = - C_{1} + C_{2} + 2 \int \limits ^{- C_{1} - x} f{\left (- r \right )}\, dr \]