Internal problem ID [9852]
Internal file name [OUTPUT/8797_Monday_June_06_2022_05_30_22_AM_27836754/index.tex]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1528.
ODE order: 3.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_3rd_order, _missing_y]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime \prime } \sin \left (x \right )+\left (2 \cos \left (x \right )+1\right ) y^{\prime \prime }-y^{\prime } \sin \left (x \right )=\cos \left (x \right )} \] Since \(y\) is missing from the ode then we can use the substitution \(y^{\prime } = v \left (x \right )\) to reduce the order by one. The ODE becomes \begin {align*} v^{\prime \prime }\left (x \right ) \sin \left (x \right )+\left (2 \cos \left (x \right )+1\right ) v^{\prime }\left (x \right )-v \left (x \right ) \sin \left (x \right ) = 0 \end {align*}
Integrating both sides of the ODE w.r.t \(x\) gives \begin {align*} \int \left (v^{\prime \prime }\left (x \right ) \sin \left (x \right )+\left (2 \cos \left (x \right )+1\right ) v^{\prime }\left (x \right )-v \left (x \right ) \sin \left (x \right )\right )d x &= 0 \\ \left (\cos \left (x \right )+1\right ) v \left (x \right )+v^{\prime }\left (x \right ) \sin \left (x \right ) = c_{1} \end {align*}
Which is now solved for \(v \left (x \right )\).
Entering Linear first order ODE solver. In canonical form a linear first order is \begin {align*} v^{\prime }\left (x \right ) + p(x)v \left (x \right ) &= q(x) \end {align*}
Where here \begin {align*} p(x) &=\frac {\cos \left (x \right )+1}{\sin \left (x \right )}\\ q(x) &=\frac {c_{1}}{\sin \left (x \right )} \end {align*}
Hence the ode is \begin {align*} v^{\prime }\left (x \right )+\frac {\left (\cos \left (x \right )+1\right ) v \left (x \right )}{\sin \left (x \right )} = \frac {c_{1}}{\sin \left (x \right )} \end {align*}
The integrating factor \(\mu \) is \begin{align*} \mu &= {\mathrm e}^{\int \frac {\cos \left (x \right )+1}{\sin \left (x \right )}d x} \\ &= {\mathrm e}^{\ln \left (\sin \left (x \right )\right )+\ln \left (\csc \left (x \right )-\cot \left (x \right )\right )} \\ \end{align*} Which simplifies to \[ \mu = 1-\cos \left (x \right ) \] The ode becomes \begin {align*} \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}}\left ( \mu v\right ) &= \left (\mu \right ) \left (\frac {c_{1}}{\sin \left (x \right )}\right ) \\ \frac {\mathop {\mathrm {d}}}{ \mathop {\mathrm {d}x}} \left (\left (1-\cos \left (x \right )\right ) v\right ) &= \left (1-\cos \left (x \right )\right ) \left (\frac {c_{1}}{\sin \left (x \right )}\right )\\ \mathrm {d} \left (\left (1-\cos \left (x \right )\right ) v\right ) &= \left (c_{1} \left (\csc \left (x \right )-\cot \left (x \right )\right )\right )\, \mathrm {d} x \end {align*}
Integrating gives \begin {align*} \left (1-\cos \left (x \right )\right ) v &= \int {c_{1} \left (\csc \left (x \right )-\cot \left (x \right )\right )\,\mathrm {d} x}\\ \left (1-\cos \left (x \right )\right ) v &= c_{1} \left (-\ln \left (\sin \left (x \right )\right )-\ln \left (\cot \left (x \right )+\csc \left (x \right )\right )\right ) + c_{2} \end {align*}
Dividing both sides by the integrating factor \(\mu =1-\cos \left (x \right )\) results in \begin {align*} v \left (x \right ) &= \frac {c_{1} \left (-\ln \left (\sin \left (x \right )\right )-\ln \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )}{1-\cos \left (x \right )}+\frac {c_{2}}{1-\cos \left (x \right )} \end {align*}
which simplifies to \begin {align*} v \left (x \right ) &= \frac {c_{1} \left (-\ln \left (\sin \left (x \right )\right )-\ln \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )+c_{2}}{1-\cos \left (x \right )} \end {align*}
But since \(y^{\prime } = v \left (x \right )\) then we now need to solve the ode \(y^{\prime } = \frac {c_{1} \left (-\ln \left (\sin \left (x \right )\right )-\ln \left (\cot \left (x \right )+\csc \left (x \right )\right )\right )+c_{2}}{1-\cos \left (x \right )}\). Integrating both sides gives \begin {align*} y = \int \frac {c_{1} \ln \left (\sin \left (x \right )\right )+c_{1} \ln \left (\cot \left (x \right )+\csc \left (x \right )\right )-c_{2}}{\cos \left (x \right )-1}d x +c_{3} \end {align*}
This is higher order nonhomogeneous ODE. Let the solution be \[ y = y_h + y_p \] Where \(y_h\) is the solution to the homogeneous ODE And \(y_p\) is a particular solution to the nonhomogeneous ODE. \(y_h\) is the solution to \[ y^{\prime \prime \prime } \sin \left (x \right )+\left (2 \cos \left (x \right )+1\right ) y^{\prime \prime }-y^{\prime } \sin \left (x \right ) = 0 \] Let the particular solution be \[ y_p = U_1 y_1+U_2 y_2+U_3 y_3 \] Where \(y_i\) are the basis solutions found above for the homogeneous solution \(y_h\) and \(U_i(x)\) are functions to be determined as follows \[ U_i = (-1)^{n-i} \int { \frac {F(x) W_i(x) }{a W(x)} \, dx} \] Where \(W(x)\) is the Wronskian and \(W_i(x)\) is the Wronskian that results after deleting the last row and the \(i\)-th column of the determinant and \(n\) is the order of the ODE or equivalently, the number of basis solutions, and \(a\) is the coefficient of the leading derivative in the ODE, and \(F(x)\) is the RHS of the ODE. Therefore, the first step is to find the Wronskian \(W \left (x \right )\). This is given by \begin {equation*} W(x) = \begin {vmatrix} y_1&y_2&y_3\\ y_1'&y_2'&y_3'\\ y_1''&y_2''&y_3''\\ \end {vmatrix} \end {equation*}
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left (\frac {d}{d x}y^{\prime \prime }\right ) \sin \left (x \right )+\left (2 \cos \left (x \right )+1\right ) \left (\frac {d}{d x}y^{\prime }\right )-y^{\prime } \sin \left (x \right )=\cos \left (x \right ) \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 3 \\ {} & {} & \frac {d}{d x}y^{\prime \prime } \end {array} \]
Maple trace
`Methods for third order ODEs: --- Trying classification methods --- trying a quadrature trying high order exact linear fully integrable <- high order exact linear fully integrable successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 68
dsolve(diff(diff(diff(y(x),x),x),x)*sin(x)+(2*cos(x)+1)*diff(diff(y(x),x),x)-diff(y(x),x)*sin(x)-cos(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = \frac {\ln \left (\csc \left (x \right )-\cot \left (x \right )\right ) c_{1} -\ln \left (\sin \left (x \right )\right ) c_{1} -\cot \left (x \right )^{2} x +\left (c_{1} x +c_{2} +1\right ) \cot \left (x \right )+\csc \left (x \right )^{2} x +\left (-c_{1} x -c_{2} -1\right ) \csc \left (x \right )-c_{3}}{-\csc \left (x \right )+\cot \left (x \right )} \]
✓ Solution by Mathematica
Time used: 4.252 (sec). Leaf size: 56
DSolve[-Cos[x] - Sin[x]*y'[x] + (1 + 2*Cos[x])*y''[x] + Sin[x]*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to \cot \left (\frac {x}{2}\right ) \arcsin (\cos (x))-\frac {c_2 x}{\sqrt {2}}-\frac {\cot \left (\frac {x}{2}\right ) (c_2 \log (2 (\cos (x)+1))+2 c_1)}{\sqrt {2}}+c_3 \]