Internal problem ID [5098]
Internal file name [OUTPUT/4591_Sunday_June_05_2022_03_01_27_PM_55050272/index.tex]
Book: Engineering Mathematics. By K. A. Stroud. 5th edition. Industrial press Inc. NY.
2001
Section: Program 24. First order differential equations. Further problems 24. page
1068
Problem number: 12.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }+y \tan \left (x \right )=\sin \left (x \right )} \]
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) &=\tan \left (x \right )\\ q(x) &=\sin \left (x \right ) \end {align*}
Hence the ode is \begin {align*} y^{\prime }+y \tan \left (x \right ) = \sin \left (x \right ) \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \tan \left (x \right )d x} \\ &= \frac {1}{\cos \left (x \right )} \\ \end{align*} Which simplifies to \[ \mu = \sec \left (x \right ) \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\sin \left (x \right )\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\sec \left (x \right ) y\right ) &= \left (\sec \left (x \right )\right ) \left (\sin \left (x \right )\right )\\ \mathrm {d} \left (\sec \left (x \right ) y\right ) &= \tan \left (x \right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \sec \left (x \right ) y &= \int {\tan \left (x \right )\,\mathrm {d} x}\\ \sec \left (x \right ) y &= -\ln \left (\cos \left (x \right )\right ) + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\sec \left (x \right )\) results in \begin {align*} y &= -\cos \left (x \right ) \ln \left (\cos \left (x \right )\right )+c_{1} \cos \left (x \right ) \end {align*}
which simplifies to \begin {align*} y &= \cos \left (x \right ) \left (-\ln \left (\cos \left (x \right )\right )+c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \cos \left (x \right ) \left (-\ln \left (\cos \left (x \right )\right )+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \cos \left (x \right ) \left (-\ln \left (\cos \left (x \right )\right )+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y \tan \left (x \right )=\sin \left (x \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 }=-y \tan \left (x \right )+\sin \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 }+y \tan \left (x \right )=\sin \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 }+y \tan \left (x \right )\right )=\mu \left (x \right ) \sin \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 }+y \tan \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 )=\mu \left (x \right ) \tan \left (x \right ) \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\frac {1}{\cos \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 ) \sin \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 ) \sin \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right ) \sin \left (x \right )d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\frac {1}{\cos \left (x \right )} \\ {} & {} & y=\cos \left (x \right ) \left (\int \frac {\sin \left (x \right )}{\cos \left (x \right )}d x +c_{1} \right ) \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\cos \left (x \right ) \left (-\ln \left (\cos \left (x \right )\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: 0.0 (sec). Leaf size: 14
dsolve(diff(y(x),x)+y(x)*tan(x)=sin(x),y(x), singsol=all)
\[ y \left (x \right ) = \left (-\ln \left (\cos \left (x \right )\right )+c_{1} \right ) \cos \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.058 (sec). Leaf size: 16
DSolve[y'[x]+y[x]*Tan[x]==Sin[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \cos (x) (-\log (\cos (x))+c_1) \]