problem 5

Internal problem ID [4765]
Internal ﬁle name [OUTPUT/4258_Sunday_June_05_2022_12_49_00_PM_92327569/index.tex]

Book: Mathematical Methods in the Physical Sciences. third edition. Mary L. Boas. John Wiley. 2006
Section: Chapter 8, Ordinary diﬀerential equations. Section 3. Linear First-Order Equations. page 403
Problem number: 5.
ODE order: 1.
ODE degree: 1.

3.5.1 Solving as linear ode

Entering Linear ﬁrst order ODE solver. In canonical form a linear ﬁrst order is \begin {align*} y^{\prime } + p(x)y &= q(x) \end {align*}

Where here \begin {align*} p(x) &=\frac {1}{\cos \left (x \right )}\\ q(x) &=\cos \left (x \right ) \end {align*}

Hence the ode is \begin {align*} y^{\prime }+\frac {y}{\cos \left (x \right )} = \cos \left (x \right ) \end {align*}

The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {1}{\cos \left (x \right )}d x} \\ &= \sec \left (x \right )+\tan \left (x \right ) \\ \end{align*} The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\cos \left (x \right )\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\left (\sec \left (x \right )+\tan \left (x \right )\right ) y\right ) &= \left (\sec \left (x \right )+\tan \left (x \right )\right ) \left (\cos \left (x \right )\right )\\ \mathrm {d} \left (\left (\sec \left (x \right )+\tan \left (x \right )\right ) y\right ) &= \left (1+\sin \left (x \right )\right )\, \mathrm {d} x \end {align*}

Integrating gives \begin {align*} \left (\sec \left (x \right )+\tan \left (x \right )\right ) y &= \int {1+\sin \left (x \right )\,\mathrm {d} x}\\ \left (\sec \left (x \right )+\tan \left (x \right )\right ) y &= x -\cos \left (x \right ) + c_{1} \end {align*}

Dividing both sides by the integrating factor \(\mu =\sec \left (x \right )+\tan \left (x \right )\) results in \begin {align*} y &= \frac {x -\cos \left (x \right )}{\sec \left (x \right )+\tan \left (x \right )}+\frac {c_{1}}{\sec \left (x \right )+\tan \left (x \right )} \end {align*}

Summary

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

Veriﬁcation of solutions

\[ y = \frac {x -\cos \left (x \right )}{\sec \left (x \right )+\tan \left (x \right )}+\frac {c_{1}}{\sec \left (x \right )+\tan \left (x \right )} \] Veriﬁed OK.

3.5.2 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime } \cos \left (x \right )+y=\cos \left (x \right )^{2} \\ \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 }=\frac {-y+\cos \left (x \right )^{2}}{\cos \left (x \right )} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=-\frac {y}{\cos \left (x \right )}+\cos \left (x \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} \\ {} & {} & y^{\prime }+\frac {y}{\cos \left (x \right )}=\cos \left (x \right ) \\ \bullet & {} & \textrm {The ODE is linear; multiply by an integrating factor}\hspace {3pt} \mu \left (x \right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+\frac {y}{\cos \left (x \right )}\right )=\mu \left (x \right ) \cos \left (x \right ) \\ \bullet & {} & \textrm {Assume the lhs of the ODE is the total derivative}\hspace {3pt} \frac {d}{d x}\left (y \mu \left (x \right )\right ) \\ {} & {} & \mu \left (x \right ) \left (y^{\prime }+\frac {y}{\cos \left (x \right )}\right )=y^{\prime } \mu \left (x \right )+y \mu ^{\prime }\left (x \right ) \\ \bullet & {} & \textrm {Isolate}\hspace {3pt} \mu ^{\prime }\left (x \right ) \\ {} & {} & \mu ^{\prime }\left (x \right )=\frac {\mu \left (x \right )}{\cos \left (x \right )} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\sec \left (x \right )+\tan \left (x \right ) \\ \bullet & {} & \textrm {Integrate both sides with respect to}\hspace {3pt} x \\ {} & {} & \int \left (\frac {d}{d x}\left (y \mu \left (x \right )\right )\right )d x =\int \mu \left (x \right ) \cos \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int \mu \left (x \right ) \cos \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right ) \cos \left (x \right )d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\sec \left (x \right )+\tan \left (x \right ) \\ {} & {} & y=\frac {\int \cos \left (x \right ) \left (\sec \left (x \right )+\tan \left (x \right )\right )d x +c_{1}}{\sec \left (x \right )+\tan \left (x \right )} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {x -\cos \left (x \right )+c_{1}}{\sec \left (x \right )+\tan \left (x \right )} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\frac {\left (x -\cos \left (x \right )+c_{1} \right ) \left (\cos \left (x \right )-\sin \left (x \right )+1\right )}{\cos \left (x \right )+1+\sin \left (x \right )} \end {array} \]

Maple trace

`Methods for first order ODEs: 
--- Trying classification methods --- 
trying a quadrature 
trying 1st order linear 
<- 1st order linear successful`

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

\[ y(x)\to e^{-2 \text {arctanh}\left (\tan \left (\frac {x}{2}\right )\right )} (x-\cos (x)+c_1) \]

3.5 problem 5

3.5.1 Solving as linear ode

3.5.2 Maple step by step solution