1.5 problem 5.1 (v)

Internal problem ID [11971]
Internal file name [OUTPUT/10624_Saturday_September_02_2023_02_48_36_PM_68303959/index.tex]

Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C. ROBINSON. Cambridge University Press 2004
Section: Chapter 5, Trivial differential equations. Exercises page 33
Problem number: 5.1 (v).
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]

\[ \boxed {T^{\prime }={\mathrm e}^{-t} \sin \left (2 t \right )} \]

1.5.1 Solving as quadrature ode

Integrating both sides gives \begin {align*} T &= \int { {\mathrm e}^{-t} \sin \left (2 t \right )\,\mathop {\mathrm {d}t}}\\ &= -\frac {\left (\sin \left (2 t \right )+2 \cos \left (2 t \right )\right ) {\mathrm e}^{-t}}{5}+c_{1} \end {align*}

Summary

The solution(s) found are the following \begin{align*} \tag{1} T &= -\frac {\left (\sin \left (2 t \right )+2 \cos \left (2 t \right )\right ) {\mathrm e}^{-t}}{5}+c_{1} \\ \end{align*}

Figure 5: Slope field plot

Verification of solutions

\[ T = -\frac {\left (\sin \left (2 t \right )+2 \cos \left (2 t \right )\right ) {\mathrm e}^{-t}}{5}+c_{1} \] Verified OK.

1.5.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & T^{\prime }={\mathrm e}^{-t} \sin \left (2 t \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & T^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int T^{\prime }d t =\int {\mathrm e}^{-t} \sin \left (2 t \right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & T=-\frac {2 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{5}-\frac {{\mathrm e}^{-t} \sin \left (2 t \right )}{5}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} T \\ {} & {} & T=-\frac {2 \,{\mathrm e}^{-t} \cos \left (2 t \right )}{5}-\frac {{\mathrm e}^{-t} \sin \left (2 t \right )}{5}+c_{1} \end {array} \]

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: 25

dsolve(diff(T(t),t)=exp(-t)*sin(2*t),T(t), singsol=all)
 

\[ T \left (t \right ) = \frac {{\mathrm e}^{-t} \left (-2 \cos \left (2 t \right )-\sin \left (2 t \right )\right )}{5}+c_{1} \]

✓ Solution by Mathematica

Time used: 0.055 (sec). Leaf size: 28

DSolve[T'[t]==Exp[-t]*Sin[2*t],T[t],t,IncludeSingularSolutions -> True]
 

\[ T(t)\to -\frac {1}{5} e^{-t} (\sin (2 t)+2 \cos (2 t))+c_1 \]