Internal problem ID [6166]
Internal file name [OUTPUT/5414_Sunday_June_05_2022_03_36_37_PM_36928189/index.tex]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven
Krantz. McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 1. What is a differential equation. Section 1.4 First Order Linear Equations.
Page 15
Problem number: 1(j).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y+x y \cot \left (x \right )+y^{\prime } x=x} \]
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 {\cot \left (x \right ) x +1}{x}\\ q(x) &=1 \end {align*}
Hence the ode is \begin {align*} y^{\prime }+\frac {\left (\cot \left (x \right ) x +1\right ) y}{x} = 1 \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {\cot \left (x \right ) x +1}{x}d x} \\ &= {\mathrm e}^{\ln \left (\sin \left (x \right )\right )+\ln \left (x \right )} \\ \end{align*} Which simplifies to \[ \mu = x \sin \left (x \right ) \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu y\right ) &= \mu \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (x \sin \left (x \right ) y\right ) &= x \sin \left (x \right )\\ \mathrm {d} \left (x \sin \left (x \right ) y\right ) &= x \sin \left (x \right )\mathrm {d} x \end {align*}
Integrating gives \begin {align*} x \sin \left (x \right ) y &= \int {x \sin \left (x \right )\,\mathrm {d} x}\\ x \sin \left (x \right ) y &= -x \cos \left (x \right )+\sin \left (x \right ) + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =x \sin \left (x \right )\) results in \begin {align*} y &= \frac {\csc \left (x \right ) \left (-x \cos \left (x \right )+\sin \left (x \right )\right )}{x}+\frac {c_{1} \csc \left (x \right )}{x} \end {align*}
which simplifies to \begin {align*} y &= \frac {-\cot \left (x \right ) x +1+c_{1} \csc \left (x \right )}{x} \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {-\cot \left (x \right ) x +1+c_{1} \csc \left (x \right )}{x} \\ \end{align*}
Verification of solutions
\[ y = \frac {-\cot \left (x \right ) x +1+c_{1} \csc \left (x \right )}{x} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y+x y \cot \left (x \right )+y^{\prime } x =x \\ \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+x -x y \cot \left (x \right )}{x} \\ \bullet & {} & \textrm {Collect w.r.t.}\hspace {3pt} y\hspace {3pt}\textrm {and simplify}\hspace {3pt} \\ {} & {} & y^{\prime }=1-\frac {\left (\cot \left (x \right ) x +1\right ) y}{x} \\ \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 {\left (\cot \left (x \right ) x +1\right ) y}{x}=1 \\ \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 {\left (\cot \left (x \right ) x +1\right ) y}{x}\right )=\mu \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 {\left (\cot \left (x \right ) x +1\right ) y}{x}\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 ) \left (\cot \left (x \right ) x +1\right )}{x} \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=x \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 \mu \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 )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int \mu \left (x \right )d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=x \sin \left (x \right ) \\ {} & {} & y=\frac {\int x \sin \left (x \right )d x +c_{1}}{x \sin \left (x \right )} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {-x \cos \left (x \right )+\sin \left (x \right )+c_{1}}{x \sin \left (x \right )} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\frac {-\cot \left (x \right ) x +1+c_{1} \csc \left (x \right )}{x} \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: 19
dsolve(y(x)-x+x*y(x)*cot(x)+x*diff(y(x),x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {-\cot \left (x \right ) x +1+\csc \left (x \right ) c_{1}}{x} \]
✓ Solution by Mathematica
Time used: 0.072 (sec). Leaf size: 21
DSolve[y[x]-x+x*y[x]*Cot[x]+x*y'[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \frac {-x \cot (x)+c_1 \csc (x)+1}{x} \]