Internal problem ID [1667]
Internal file name [OUTPUT/1668_Sunday_June_05_2022_02_26_26_AM_19524921/index.tex]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 1.2. Page 9
Problem number: 23.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {\tan \left (t \right ) y+y^{\prime }=\sin \left (t \right ) \cos \left (t \right )} \]
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} y^{\prime } + p(t)y &= q(t) \end {align*}
Where here \begin {align*} p(t) &=\tan \left (t \right )\\ q(t) &=\sin \left (t \right ) \cos \left (t \right ) \end {align*}
Hence the ode is \begin {align*} \tan \left (t \right ) y+y^{\prime } = \sin \left (t \right ) \cos \left (t \right ) \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \tan \left (t \right )d t} \\ &= \frac {1}{\cos \left (t \right )} \\ \end{align*} Which simplifies to \[ \mu = \sec \left (t \right ) \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\sin \left (t \right ) \cos \left (t \right )\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}t}} \left (\sec \left (t \right ) y\right ) &= \left (\sec \left (t \right )\right ) \left (\sin \left (t \right ) \cos \left (t \right )\right )\\ \mathrm {d} \left (\sec \left (t \right ) y\right ) &= \sin \left (t \right )\, \mathrm {d} t \end {align*}
Integrating gives \begin {align*} \sec \left (t \right ) y &= \int {\sin \left (t \right )\,\mathrm {d} t}\\ \sec \left (t \right ) y &= -\cos \left (t \right ) + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\sec \left (t \right )\) results in \begin {align*} y &= -\cos \left (t \right )^{2}+c_{1} \cos \left (t \right ) \end {align*}
which simplifies to \begin {align*} y &= \cos \left (t \right ) \left (-\cos \left (t \right )+c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \cos \left (t \right ) \left (-\cos \left (t \right )+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \cos \left (t \right ) \left (-\cos \left (t \right )+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \tan \left (t \right ) y+y^{\prime }=\sin \left (t \right ) \cos \left (t \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=-\tan \left (t \right ) y+\sin \left (t \right ) \cos \left (t \right ) \\ \bullet & {} & \textrm {Group terms with}\hspace {3pt} y\hspace {3pt}\textrm {on the lhs of the ODE and the rest on the rhs of the ODE}\hspace {3pt} \\ {} & {} & \tan \left (t \right ) y+y^{\prime }=\sin \left (t \right ) \cos \left (t \right ) \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (t \right ) \\ {} & {} & \mu \left (t \right ) \left (\tan \left (t \right ) y+y^{\prime }\right )=\mu \left (t \right ) \sin \left (t \right ) \cos \left (t \right ) \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d t}\left (y \mu \left (t \right )\right ) \\ {} & {} & \mu \left (t \right ) \left (\tan \left (t \right ) y+y^{\prime }\right )=y^{\prime } \mu \left (t \right )+y \mu ^{\prime }\left (t \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (t \right ) \\ {} & {} & \mu ^{\prime }\left (t \right )=\mu \left (t \right ) \tan \left (t \right ) \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (t \right )=\frac {1}{\cos \left (t \right )} \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} t \\ {} & {} & \int \left (\frac {d}{d t}\left (y \mu \left (t \right )\right )\right )d t =\int \mu \left (t \right ) \sin \left (t \right ) \cos \left (t \right )d t +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (t \right )=\int \mu \left (t \right ) \sin \left (t \right ) \cos \left (t \right )d t +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (t \right ) \sin \left (t \right ) \cos \left (t \right )d t +c_{1}}{\mu \left (t \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (t \right )=\frac {1}{\cos \left (t \right )} \\ {} & {} & y=\cos \left (t \right ) \left (\int \sin \left (t \right )d t +c_{1} \right ) \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\cos \left (t \right ) \left (-\cos \left (t \right )+c_{1} \right ) \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: 0.015 (sec). Leaf size: 13
dsolve(diff(y(t),t)+tan(t)*y(t)=cos(t)*sin(t),y(t), singsol=all)
\[ y \left (t \right ) = \left (-\cos \left (t \right )+c_{1} \right ) \cos \left (t \right ) \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 15
DSolve[y'[t]+Tan[t]*y[t]==Cos[t]*Sin[t],y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to \cos (t) (-\cos (t)+c_1) \]