Internal problem ID [14472]
Internal file name [OUTPUT/14153_Monday_March_25_2024_09_50_30_PM_24630254/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.1, page 141
Problem number: 47.
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]]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime \prime }+b \left (t \right ) y^{\prime }+c \left (t \right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\cos \left (t \right )}{t^{3}} \end {align*}
With initial conditions \begin {align*} [y \left (\pi \right ) = 0, y^{\prime }\left (2 \pi \right ) = 0] \end {align*}
Given one basis solution \(y_{1}\left (t \right )\), then the second basis solution is given by \[ y_{2}\left (t \right ) = y_{1} \left (\int \frac {{\mathrm e}^{-\left (\int p d t \right )}}{y_{1}^{2}}d t \right ) \] Where \(p(x)\) is the coefficient of \(y^{\prime }\) when the ode is written in the normal form \[ y^{\prime \prime }+p \left (t \right ) y^{\prime }+q \left (t \right ) y = f \left (t \right ) \] Looking at the ode to solve shows that \[ p \left (t \right ) = b \left (t \right ) \] Therefore \begin{align*} y_{2}\left (t \right ) &= \frac {\cos \left (t \right ) \left (\int \frac {{\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}}{\cos \left (t \right )^{2}}d t \right )}{t^{3}} \\ y_{2}\left (t \right ) &= \frac {\cos \left (t \right )}{t^{3}} \int \frac {{\mathrm e}^{-\left (\int b \left (t \right )d t \right )}}{\frac {\cos \left (t \right )^{2}}{t^{6}}} , dt \\ y_{2}\left (t \right ) &= \frac {\cos \left (t \right ) \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )}{t^{3}} \\ y_{2}\left (t \right ) &= \frac {\cos \left (t \right ) \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )}{t^{3}} \\ \end{align*} Hence the solution is \begin{align*} y &= c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ &= \frac {c_{1} \cos \left (t \right )}{t^{3}}+\frac {c_{2} \cos \left (t \right ) \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )}{t^{3}} \\ \end{align*} Initial conditions are used to solve for the constants of integration.
Looking at the above solution \begin {align*} y = \frac {c_{1} \cos \left (t \right )}{t^{3}}+\frac {c_{2} \cos \left (t \right ) \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )}{t^{3}} \tag {1} \end {align*}
Initial conditions are now substituted in the above solution. This will generate the required equations to solve for the integration constants. substituting \(y = 0\) and \(t = \pi \) in the above gives \begin {align*} 0 = \munderset {t \rightarrow \pi }{\operatorname {lim}}\frac {\cos \left (t \right ) \left (c_{1} +c_{2} \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )\right )}{t^{3}}\tag {1A} \end {align*}
Taking derivative of the solution gives \begin {align*} y^{\prime } = -\frac {3 c_{1} \cos \left (t \right )}{t^{4}}-\frac {c_{1} \sin \left (t \right )}{t^{3}}-\frac {3 c_{2} \cos \left (t \right ) \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )}{t^{4}}-\frac {c_{2} \sin \left (t \right ) \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )}{t^{3}}+c_{2} t^{3} \cos \left (t \right ) \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} \end {align*}
substituting \(y^{\prime } = 0\) and \(t = 2 \pi \) in the above gives \begin {align*} 0 = -\left (\munderset {t \rightarrow 2 \pi }{\operatorname {lim}}\frac {c_{2} \left (t \sin \left (t \right )+3 \cos \left (t \right )\right ) \left (\int \sec \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{6}d t \right )-c_{2} t^{7} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} \sec \left (t \right )+c_{1} \left (t \sin \left (t \right )+3 \cos \left (t \right )\right )}{t^{4}}\right )\tag {2A} \end {align*}
Equations {1A,2A} are now solved for \(\{c_{1}, c_{2}\}\). There is no solution for the constants of integrations. This solution is removed.
Verification of solutions N/A
✗ Solution by Maple
dsolve([diff(diff(y(t),t),t)+b(t)*diff(y(t),t)+c(t)*y(t) = 0, 1/t^3*cos(t), y(Pi) = 0, D(y)(2*Pi) = 0], singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[t]+b[t]*y'[t]+c[t]*y[t]==0,{y[Pi]==0,y'[2*Pi]==0},y[t],t,IncludeSingularSolutions -> True]
Not solved