Internal problem ID [3198]
Internal file name [OUTPUT/2690_Sunday_June_05_2022_08_38_51_AM_15825738/index.tex]
Book: Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section: Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page
78
Problem number: 54.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y \cos \left (x \right )-\sin \left (x \right ) y^{\prime }=-1} \]
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 {\cos \left (x \right )}{\sin \left (x \right )}\\ q(x) &=\frac {1}{\sin \left (x \right )} \end {align*}
Hence the ode is \begin {align*} y^{\prime }-\frac {\cos \left (x \right ) y}{\sin \left (x \right )} = \frac {1}{\sin \left (x \right )} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int -\frac {\cos \left (x \right )}{\sin \left (x \right )}d x} \\ &= \frac {1}{\sin \left (x \right )} \\ \end{align*} Which simplifies to \[ \mu = \csc \left (x \right ) \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \left (\mu \right ) \left (\frac {1}{\sin \left (x \right )}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\csc \left (x \right ) y\right ) &= \left (\csc \left (x \right )\right ) \left (\frac {1}{\sin \left (x \right )}\right )\\ \mathrm {d} \left (\csc \left (x \right ) y\right ) &= \csc \left (x \right )^{2}\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \csc \left (x \right ) y &= \int {\csc \left (x \right )^{2}\,\mathrm {d} x}\\ \csc \left (x \right ) y &= -\cot \left (x \right ) + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\csc \left (x \right )\) results in \begin {align*} y &= -\sin \left (x \right ) \cot \left (x \right )+c_{1} \sin \left (x \right ) \end {align*}
which simplifies to \begin {align*} y &= c_{1} \sin \left (x \right )-\cos \left (x \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= c_{1} \sin \left (x \right )-\cos \left (x \right ) \\ \end{align*}
Verification of solutions
\[ y = c_{1} \sin \left (x \right )-\cos \left (x \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y \cos \left (x \right )-\sin \left (x \right ) y^{\prime }=-1 \\ \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 {-1-y \cos \left (x \right )}{\sin \left (x \right )} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=\frac {\cos \left (x \right ) y}{\sin \left (x \right )}+\frac {1}{\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 }-\frac {\cos \left (x \right ) y}{\sin \left (x \right )}=\frac {1}{\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 }-\frac {\cos \left (x \right ) y}{\sin \left (x \right )}\right )=\frac {\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 }-\frac {\cos \left (x \right ) y}{\sin \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 )}{\sin \left (x \right )} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\frac {1}{\sin \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 \frac {\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 \frac {\mu \left (x \right )}{\sin \left (x \right )}d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \frac {\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}{\sin \left (x \right )} \\ {} & {} & y=\sin \left (x \right ) \left (\int \frac {1}{\sin \left (x \right )^{2}}d x +c_{1} \right ) \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\sin \left (x \right ) \left (-\cot \left (x \right )+c_{1} \right ) \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=c_{1} \sin \left (x \right )-\cos \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`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 13
dsolve((1+y(x)*cos(x))-( sin(x))*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \sin \left (x \right )-\cos \left (x \right ) \]
✓ Solution by Mathematica
Time used: 0.042 (sec). Leaf size: 15
DSolve[(1+y[x]*Cos[x])-( Sin[x])*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to -\cos (x)+c_1 \sin (x) \]