Internal problem ID [5748]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [_separable]
\[ \boxed {\left (1+y^{2}\right ) \left ({\mathrm e}^{2 x}-y^{\prime } {\mathrm e}^{y}\right )-\left (1+y\right ) y^{\prime }=0} \]
✓ Solution by Maple
Time used: 0.093 (sec). Leaf size: 30
dsolve((1+y(x)^2)*(exp(2*x)-exp(y(x))*diff(y(x),x))-(1+y(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ \frac {{\mathrm e}^{2 x}}{2}-\arctan \left (y \left (x \right )\right )-\frac {\ln \left (1+y \left (x \right )^{2}\right )}{2}-{\mathrm e}^{y \left (x \right )}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 0.696 (sec). Leaf size: 70
DSolve[(1+y[x]^2)*(Exp[2*x]-Exp[y[x]]*y'[x])-(1+y[x])*y'[x]==0,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*}