Internal problem ID [11362]
Internal file name [OUTPUT/10345_Wednesday_May_17_2023_07_49_35_PM_51119002/index.tex]
Book: A First Course in Differential Equations by J. David Logan. Third Edition. Springer-Verlag,
NY. 2015.
Section: Chapter 1, First order differential equations. Section 1.2 Antiderivatives. Exercises page
19
Problem number: 4(a).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "quadrature"
Maple gives the following as the ode type
[_quadrature]
\[ \boxed {x^{\prime }=t \,{\mathrm e}^{-2 t}} \]
Integrating both sides gives \begin {align*} x &= \int { t \,{\mathrm e}^{-2 t}\,\mathop {\mathrm {d}t}}\\ &= \left (-\frac {t}{2}-\frac {1}{4}\right ) {\mathrm e}^{-2 t}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= \left (-\frac {t}{2}-\frac {1}{4}\right ) {\mathrm e}^{-2 t}+c_{1} \\ \end{align*}
Verification of solutions
\[ x = \left (-\frac {t}{2}-\frac {1}{4}\right ) {\mathrm e}^{-2 t}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }=t \,{\mathrm e}^{-2 t} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int x^{\prime }d t =\int t \,{\mathrm e}^{-2 t}d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x=-\frac {\left (1+2 t \right ) {\mathrm e}^{-2 t}}{4}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=-\frac {t \,{\mathrm e}^{-2 t}}{2}-\frac {{\mathrm e}^{-2 t}}{4}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve(diff(x(t),t)=t*exp(-2*t),x(t), singsol=all)
\[ x \left (t \right ) = \frac {\left (-2 t -1\right ) {\mathrm e}^{-2 t}}{4}+c_{1} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 22
DSolve[x'[t]==t*Exp[-2*t],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to -\frac {1}{4} e^{-2 t} (2 t+1)+c_1 \]