Internal problem ID [12052]
Internal file name [OUTPUT/10705_Sunday_September_03_2023_12_36_43_PM_33139359/index.tex]
Book: AN INTRODUCTION TO ORDINARY DIFFERENTIAL EQUATIONS by JAMES C.
ROBINSON. Cambridge University Press 2004
Section: Chapter 17, Reduction of order. Exercises page 162
Problem number: 17.6.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "reduction_of_order"
Maple gives the following as the ode type
[[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\tan \left (t \right ) x^{\prime \prime }-3 x^{\prime }+\left (\tan \left (t \right )+3 \cot \left (t \right )\right ) x=0} \] Given that one solution of the ode is \begin {align*} x_1 &= \sin \left (t \right ) \end {align*}
Given one basis solution \(x_{1}\left (t \right )\), then the second basis solution is given by \[ x_{2}\left (t \right ) = x_{1} \left (\int \frac {{\mathrm e}^{-\left (\int p d t \right )}}{x_{1}^{2}}d t \right ) \] Where \(p(x)\) is the coefficient of \(x^{\prime }\) when the ode is written in the normal form \[ x^{\prime \prime }+p \left (t \right ) x^{\prime }+q \left (t \right ) x = f \left (t \right ) \] Looking at the ode to solve shows that \[ p \left (t \right ) = -\frac {3}{\tan \left (t \right )} \] Therefore \begin{align*} x_{2}\left (t \right ) &= \sin \left (t \right ) \left (\int \frac {{\mathrm e}^{-\left (\int -\frac {3}{\tan \left (t \right )}d t \right )}}{\sin \left (t \right )^{2}}d t \right ) \\ x_{2}\left (t \right ) &= \sin \left (t \right ) \int \frac {\sin \left (t \right )^{3}}{\sin \left (t \right )^{2}} , dt \\ x_{2}\left (t \right ) &= \sin \left (t \right ) \left (\int \sin \left (t \right )d t \right ) \\ x_{2}\left (t \right ) &= -\sin \left (t \right ) \cos \left (t \right ) \\ \end{align*} Hence the solution is \begin{align*} x &= c_{1} x_{1}\left (t \right )+c_{2} x_{2}\left (t \right ) \\ &= \sin \left (t \right ) c_{1} -c_{2} \sin \left (t \right ) \cos \left (t \right ) \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} x &= \sin \left (t \right ) c_{1} -c_{2} \sin \left (t \right ) \cos \left (t \right ) \\ \end{align*}
Verification of solutions
\[ x = \sin \left (t \right ) c_{1} -c_{2} \sin \left (t \right ) \cos \left (t \right ) \] Verified OK.
Maple trace Kovacic algorithm successful
`Methods for second order ODEs: --- Trying classification methods --- trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Heun: Equivalence to the GHE or one of its 4 confluent cases under a power @ Moebius -> trying a solution of the form r0(x) * Y + r1(x) * Y where Y = exp(int(r(x), dx)) * 2F1([a1, a2], [b1], f) -> Trying changes of variables to rationalize or make the ODE simpler trying a quadrature checking if the LODE has constant coefficients checking if the LODE is of Euler type trying a symmetry of the form [xi=0, eta=F(x)] checking if the LODE is missing y -> Trying a Liouvillian solution using Kovacics algorithm A Liouvillian solution exists Reducible group (found an exponential solution) Reducible group (found another exponential solution) <- Kovacics algorithm successful Change of variables used: [t = arcsin(t)] Linear ODE actually solved: (-2*t^2+3)*u(t)+(2*t^3-3*t)*diff(u(t),t)+(-t^4+t^2)*diff(diff(u(t),t),t) = 0 <- change of variables successful`
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 13
dsolve([tan(t)*diff(x(t),t$2)-3*diff(x(t),t)+(tan(t)+3*cot(t))*x(t)=0,sin(t)],singsol=all)
\[ x \left (t \right ) = \sin \left (t \right ) \left (c_{1} +c_{2} \cos \left (t \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.374 (sec). Leaf size: 24
DSolve[Tan[t]*x''[t]-3*x'[t]+(Tan[t]+3*Cot[t])*x[t]==0,x[t],t,IncludeSingularSolutions -> True]
\[ x(t)\to \sqrt {-\sin ^2(t)} (c_2 \cos (t)+c_1) \]