Internal problem ID [1881]
Internal file name [OUTPUT/1882_Sunday_June_05_2022_02_36_41_AM_32393311/index.tex]
Book: Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath.
Boston. 1964
Section: Exercise 5, page 21
Problem number: 12.
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]
\[ \boxed {x^{\prime }=1-\sin \left (2 t \right )} \]
Integrating both sides gives \begin {align*} x &= \int { 1-\sin \left (2 t \right )\,\mathop {\mathrm {d}t}}\\ &= t +\frac {\cos \left (2 t \right )}{2}+c_{1} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} x &= t +\frac {\cos \left (2 t \right )}{2}+c_{1} \\ \end{align*}
Verification of solutions
\[ x = t +\frac {\cos \left (2 t \right )}{2}+c_{1} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & x^{\prime }=1-\sin \left (2 t \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & x^{\prime } \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int x^{\prime }d t =\int \left (1-\sin \left (2 t \right )\right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate integral}\hspace {3pt} \\ {} & {} & x=t +\frac {\cos \left (2 t \right )}{2}+c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} x \\ {} & {} & x=t +\frac {\cos \left (2 t \right )}{2}+c_{1} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature <- quadrature successful`
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 13
dsolve(diff(x(t),t)=1-sin(2*t),x(t), singsol=all)
\[ x \left (t \right ) = \frac {\cos \left (2 t \right )}{2}+t +c_{1} \]
✓ Solution by Mathematica
Time used: 0.022 (sec). Leaf size: 17
DSolve[x'[t]==1-Sin[2*t],x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to t+\frac {1}{2} \cos (2 t)+c_1 \]