Internal problem ID [14471]
Internal file name [OUTPUT/14152_Monday_March_25_2024_09_50_29_PM_43243319/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: 46.
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]]
\[ \boxed {y^{\prime \prime }+b \left (t \right ) y^{\prime }+c \left (t \right ) y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \left (t \right )}{t^{2}} \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 {\sin \left (t \right ) \left (\int \frac {{\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{4}}{\sin \left (t \right )^{2}}d t \right )}{t^{2}} \\ y_{2}\left (t \right ) &= \frac {\sin \left (t \right )}{t^{2}} \int \frac {{\mathrm e}^{-\left (\int b \left (t \right )d t \right )}}{\frac {\sin \left (t \right )^{2}}{t^{4}}} , dt \\ y_{2}\left (t \right ) &= \frac {\sin \left (t \right ) \left (\int \csc \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{4}d t \right )}{t^{2}} \\ y_{2}\left (t \right ) &= \frac {\sin \left (t \right ) \left (\int \csc \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{4}d t \right )}{t^{2}} \\ \end{align*} Hence the solution is \begin{align*} y &= c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ &= \frac {\sin \left (t \right ) c_{1}}{t^{2}}+\frac {c_{2} \sin \left (t \right ) \left (\int \csc \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{4}d t \right )}{t^{2}} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \frac {\sin \left (t \right ) c_{1}}{t^{2}}+\frac {c_{2} \sin \left (t \right ) \left (\int \csc \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{4}d t \right )}{t^{2}} \\ \end{align*}
Verification of solutions
\[ y = \frac {\sin \left (t \right ) c_{1}}{t^{2}}+\frac {c_{2} \sin \left (t \right ) \left (\int \csc \left (t \right )^{2} {\mathrm e}^{-\left (\int b \left (t \right )d t \right )} t^{4}d t \right )}{t^{2}} \] Verified OK.
✗ Solution by Maple
dsolve([diff(y(t),t$2)+b(t)*diff(y(t),t)+c(t)*y(t)=0,1/t^2*sin(t)],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[t],t,IncludeSingularSolutions -> True]
Not solved