75.10.1 problem 232

Internal problem ID [16848]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Section 9. The Riccati equation. Exercises page 75
Problem number : 232
Date solved : Tuesday, January 28, 2025 at 09:34:49 AM
CAS classification : [[_1st_order, `_with_symmetry_[F(x),G(x)]`], _Riccati]

\begin{align*} y^{\prime } {\mathrm e}^{-x}+y^{2}-2 y \,{\mathrm e}^{x}&=1-{\mathrm e}^{2 x} \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 23

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

\[ y = \frac {{\mathrm e}^{x}+{\mathrm e}^{2 x} c_{1} +c_{1}}{{\mathrm e}^{x} c_{1} +1} \]

✓ Solution by Mathematica

Time used: 0.300 (sec). Leaf size: 24

DSolve[D[y[x],x]*Exp[-x]+y[x]^2-2*y[x]*Exp[x]==1-Exp[2*x],y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to e^x+\frac {1}{e^x+c_1} \\ y(x)\to e^x \\ \end{align*}