75.4.10 problem 55

Internal problem ID [16633]
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 : 55
Date solved : Thursday, March 13, 2025 at 08:28:00 AM
CAS classification : [_separable]

\begin{align*} {\mathrm e}^{y} \left (x^{2}+1\right ) y^{\prime }-2 x \left (1+{\mathrm e}^{y}\right )&=0 \end{align*}

✓ Maple. Time used: 0.082 (sec). Leaf size: 13

ode:=exp(y(x))*(x^2+1)*diff(y(x),x)-2*x*(1+exp(y(x))) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \ln \left (c_{1} x^{2}+c_{1} -1\right ) \]

✓ Mathematica. Time used: 0.645 (sec). Leaf size: 27

ode=Exp[y[x]]*(1+x^2)*D[y[x],x]-2*x*(1+Exp[y[x]])==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \log \left (-1+e^{c_1} \left (x^2+1\right )\right ) \\ y(x)\to i \pi \\ \end{align*}

✓ Sympy. Time used: 0.308 (sec). Leaf size: 12

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*x*(exp(y(x)) + 1) + (x**2 + 1)*exp(y(x))*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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