Internal problem ID [12923]
Internal file name [OUTPUT/11576_Tuesday_November_07_2023_11_27_16_PM_37495739/index.tex]
Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th
edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.3 page 47
Problem number: 19 a(ii).
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]
\[ \boxed {\theta ^{\prime }=2} \]
Integrating both sides gives \begin {align*} \theta &= \int { 2\,\mathop {\mathrm {d}t}}\\ &= 2 t +c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} \theta &= 2 t +c_{1} \\ \end{align*}
Verification of solutions
\[ \theta = 2 t +c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \theta ^{\prime }=2 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & \theta ^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \theta ^{\prime }d t =\int 2d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & \theta =2 t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} \theta \\ {} & {} & \theta =2 t +c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear <- 1st order linear successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 9
dsolve(diff(theta(t),t)=1-cos(theta(t))+(1+cos(theta(t))),theta(t), singsol=all)
\[ \theta \left (t \right ) = 2 t +c_{1} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 11
DSolve[theta'[t]==1-Cos[theta[t]]+(1+Cos[theta[t]]),theta[t],t,IncludeSingularSolutions -> True]
\[ \theta (t)\to 2 t+c_1 \]