1.1 problem 5.1 (i)

Internal problem ID [11967]
Internal file name [OUTPUT/10620_Saturday_September_02_2023_02_48_34_PM_75656588/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 (i).
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]

\[ \boxed {x^{\prime }=\cos \left (t \right )+\sin \left (t \right )} \]

1.1.1 Solving as quadrature ode

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

Summary

The solution(s) found are the following \begin{align*} \tag{1} x &= \sin \left (t \right )-\cos \left (t \right )+c_{1} \\ \end{align*}

Figure 1: Slope field plot

Verification of solutions

\[ x = \sin \left (t \right )-\cos \left (t \right )+c_{1} \] Verified OK.

1.1.2 Maple step by step solution

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

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: 12

dsolve(diff(x(t),t)=sin(t)+cos(t),x(t), singsol=all)
 

\[ x \left (t \right ) = -\cos \left (t \right )+\sin \left (t \right )+c_{1} \]

✓ Solution by Mathematica

Time used: 0.011 (sec). Leaf size: 14

DSolve[x'[t]==Sin[t]+Cos[t],x[t],t,IncludeSingularSolutions -> True]
 

\[ x(t)\to \sin (t)-\cos (t)+c_1 \]