Internal problem ID [4995]
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]
Solve \begin {gather*} \boxed {\left (1+y^{2}\right ) \left ({\mathrm e}^{2 x}-{\mathrm e}^{y} y^{\prime }\right )-\left (y+1\right ) y^{\prime }=0} \end {gather*}
✓ Solution by Maple
Time used: 0.094 (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)
\[ \frac {{\mathrm e}^{2 x}}{2}-\arctan \left (y \relax (x )\right )-\frac {\ln \left (1+y \relax (x )^{2}\right )}{2}-{\mathrm e}^{y \relax (x )}+c_{1} = 0 \]
✓ Solution by Mathematica
Time used: 1.088 (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*}