47.1.35 problem 35

Internal problem ID [7416]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 1. First order differential equations. Section 1.1 Separable equations problems. page 7
Problem number : 35
Date solved : Sunday, March 30, 2025 at 11:58:23 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.037 (sec). Leaf size: 30

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

\[ \frac {{\mathrm e}^{2 x}}{2}-\arctan \left (y\right )-\frac {\ln \left (1+y^{2}\right )}{2}-{\mathrm e}^{y}+c_1 = 0 \]

✓ Mathematica. Time used: 0.779 (sec). Leaf size: 70

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

\begin{align*} y(x)\to \text {InverseFunction}\left [e^{\text {$\#$1}}+\left (\frac {1}{2}-\frac {i}{2}\right ) \log (-\text {$\#$1}+i)+\left (\frac {1}{2}+\frac {i}{2}\right ) \log (\text {$\#$1}+i)\&\right ]\left [\frac {e^{2 x}}{2}+c_1\right ] \\ y(x)\to -i \\ y(x)\to i \\ \end{align*}

✓ Sympy. Time used: 0.594 (sec). Leaf size: 27

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

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