71.3.20 problem 15

Internal problem ID [14299]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 2. The Initial Value Problem. Exercises 2.1, page 40
Problem number : 15
Date solved : Monday, March 31, 2025 at 12:16:06 PM
CAS classification : [[_homogeneous, `class C`], _dAlembert]

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

✓ Maple. Time used: 0.011 (sec). Leaf size: 29

ode:=diff(y(x),x) = ln(x+y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = {\mathrm e}^{\operatorname {RootOf}\left (c_1 \,{\mathrm e}-x \,{\mathrm e}-\operatorname {Ei}_{1}\left (-\textit {\_Z} -1\right )\right )}-x \]

✓ Mathematica. Time used: 0.198 (sec). Leaf size: 161

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

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

✓ Sympy. Time used: 2.037 (sec). Leaf size: 36

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

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