Internal problem ID [6165]
Internal file name [OUTPUT/5413_Sunday_June_05_2022_03_36_36_PM_27167940/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(i).
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program :
Maple gives the following as the ode type
[_linear]
\[ \boxed {y^{\prime }+y \cot \left (x \right )=2 x \csc \left (x \right )} \]
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) &=\cot \left (x \right )\\ q(x) &=2 x \csc \left (x \right ) \end {align*}
Hence the ode is \begin {align*} y^{\prime }+y \cot \left (x \right ) = 2 x \csc \left (x \right ) \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \cot \left (x \right )d x} \\ &= \sin \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 (2 x \csc \left (x \right )\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (y \sin \left (x \right )\right ) &= \left (\sin \left (x \right )\right ) \left (2 x \csc \left (x \right )\right )\\ \mathrm {d} \left (y \sin \left (x \right )\right ) &= \left (2 x\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} y \sin \left (x \right ) &= \int {2 x\,\mathrm {d} x}\\ y \sin \left (x \right ) &= x^{2} + c_{1} \end {align*}
Dividing both sides by the integrating factor \(\mu =\sin \left (x \right )\) results in \begin {align*} y &= \csc \left (x \right ) x^{2}+c_{1} \csc \left (x \right ) \end {align*}
which simplifies to \begin {align*} y &= \csc \left (x \right ) \left (x^{2}+c_{1} \right ) \end {align*}
Summary
The solution(s) found are the following \begin{align*} \tag{1} y &= \csc \left (x \right ) \left (x^{2}+c_{1} \right ) \\ \end{align*}
Verification of solutions
\[ y = \csc \left (x \right ) \left (x^{2}+c_{1} \right ) \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }+y \cot \left (x \right )=2 x \csc \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 \cot \left (x \right )+2 x \csc \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 \cot \left (x \right )=2 x \csc \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 \cot \left (x \right )\right )=2 \mu \left (x \right ) x \csc \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 \cot \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 ) \cot \left (x \right ) \\ \bullet & {} & \textrm {Solve to find the integrating factor}\hspace {3pt} \\ {} & {} & \mu \left (x \right )=\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 2 \mu \left (x \right ) x \csc \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Evaluate the integral on the lhs}\hspace {3pt} \\ {} & {} & y \mu \left (x \right )=\int 2 \mu \left (x \right ) x \csc \left (x \right )d x +c_{1} \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} y \\ {} & {} & y=\frac {\int 2 \mu \left (x \right ) x \csc \left (x \right )d x +c_{1}}{\mu \left (x \right )} \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} \mu \left (x \right )=\sin \left (x \right ) \\ {} & {} & y=\frac {\int 2 x \csc \left (x \right ) \sin \left (x \right )d x +c_{1}}{\sin \left (x \right )} \\ \bullet & {} & \textrm {Evaluate the integrals on the rhs}\hspace {3pt} \\ {} & {} & y=\frac {x^{2}+c_{1}}{\sin \left (x \right )} \\ \bullet & {} & \textrm {Simplify}\hspace {3pt} \\ {} & {} & y=\csc \left (x \right ) \left (x^{2}+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: 12
dsolve(diff(y(x),x)+y(x)*cot(x)=2*x*csc(x),y(x), singsol=all)
\[ y \left (x \right ) = \csc \left (x \right ) \left (x^{2}+c_{1} \right ) \]
✓ Solution by Mathematica
Time used: 0.048 (sec). Leaf size: 14
DSolve[y'[x]+y[x]*Cot[x]==2*x*Csc[x],y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \left (x^2+c_1\right ) \csc (x) \]