75.4.8 problem 53

Internal problem ID [16631]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 4. Equations with variables separable and equations reducible to them. Exercises page 38
Problem number : 53
Date solved : Thursday, March 13, 2025 at 08:27:55 AM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (1\right )&=1 \end{align*}

✓ Maple. Time used: 2.679 (sec). Leaf size: 22

ode:=y(x)*ln(y(x))+x*diff(y(x),x) = 1; 
ic:=y(1) = 1; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✗ Mathematica

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

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✓ Sympy. Time used: 0.471 (sec). Leaf size: 24

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

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