13.4.4 problem 6

Internal problem ID [2341]
Book : Differential equations and their applications, 3rd ed., M. Braun
Section : Section 1.9. Page 66
Problem number : 6
Date solved : Monday, January 27, 2025 at 05:48:14 AM
CAS classification : [[_1st_order, _with_linear_symmetries], [_Abel, `2nd type`, `class A`]]

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

✓ Solution by Maple

Time used: 0.036 (sec). Leaf size: 45

dsolve(1/2*y(t)^2-2*exp(t)*y(t)+(-exp(t)+y(t))*diff(y(t),t) = 0,y(t), singsol=all)
 

\begin{align*} y &= \left (1-\sqrt {\left ({\mathrm e}^{3 t}+c_1 \right ) {\mathrm e}^{-3 t}}\right ) {\mathrm e}^{t} \\ y &= \left (1+\sqrt {\left ({\mathrm e}^{3 t}+c_1 \right ) {\mathrm e}^{-3 t}}\right ) {\mathrm e}^{t} \\ \end{align*}

✓ Solution by Mathematica

Time used: 1.216 (sec). Leaf size: 70

DSolve[1/2*y[t]^2-2*Exp[t]*y[t]+(-Exp[t]+y[t])*D[y[t],t] == 0,y[t],t,IncludeSingularSolutions -> True]
 

\begin{align*} y(t)\to e^t-\frac {\sqrt {-e^{3 t}-c_1}}{\sqrt {-e^t}} \\ y(t)\to e^t+\frac {\sqrt {-e^{3 t}-c_1}}{\sqrt {-e^t}} \\ \end{align*}