Internal problem ID [14462]
Internal file name [OUTPUT/14143_Monday_March_25_2024_09_50_25_PM_63561329/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: 30.
ODE order: 2.
ODE degree: 1.
The type(s) of ODE detected by this program : "reduction_of_order", "second_order_linear_constant_coeff", "second_order_ode_can_be_made_integrable"
Maple gives the following as the ode type
[[_2nd_order, _missing_x]]
\[ \boxed {y^{\prime \prime }+49 y=0} \] Given that one solution of the ode is \begin {align*} y_1 &= \sin \left (7 t \right ) \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 \[ p \left (t \right ) = 0 \] Therefore \begin{align*} y_{2}\left (t \right ) &= \sin \left (7 t \right ) \left (\int \frac {{\mathrm e}^{-\left (\int 0d t \right )}}{\sin \left (7 t \right )^{2}}d t \right ) \\ y_{2}\left (t \right ) &= \sin \left (7 t \right ) \int \frac {1}{\sin \left (7 t \right )^{2}} , dt \\ y_{2}\left (t \right ) &= \sin \left (7 t \right ) \left (\int \csc \left (7 t \right )^{2}d t \right ) \\ y_{2}\left (t \right ) &= -\frac {\sin \left (7 t \right ) \cot \left (7 t \right )}{7} \\ \end{align*} Hence the solution is \begin{align*} y &= c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ &= \sin \left (7 t \right ) c_{1} -\frac {c_{2} \sin \left (7 t \right ) \cot \left (7 t \right )}{7} \\ \end{align*}
The solution(s) found are the following \begin{align*} \tag{1} y &= \sin \left (7 t \right ) c_{1} -\frac {c_{2} \sin \left (7 t \right ) \cot \left (7 t \right )}{7} \\ \end{align*}
Verification of solutions
\[ y = \sin \left (7 t \right ) c_{1} -\frac {c_{2} \sin \left (7 t \right ) \cot \left (7 t \right )}{7} \] Verified OK.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime \prime }+49 y=0 \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 2 \\ {} & {} & y^{\prime \prime } \\ \bullet & {} & \textrm {Characteristic polynomial of ODE}\hspace {3pt} \\ {} & {} & r^{2}+49=0 \\ \bullet & {} & \textrm {Use quadratic formula to solve for}\hspace {3pt} r \\ {} & {} & r =\frac {0\pm \left (\sqrt {-196}\right )}{2} \\ \bullet & {} & \textrm {Roots of the characteristic polynomial}\hspace {3pt} \\ {} & {} & r =\left (-7 \,\mathrm {I}, 7 \,\mathrm {I}\right ) \\ \bullet & {} & \textrm {1st solution of the ODE}\hspace {3pt} \\ {} & {} & y_{1}\left (t \right )=\cos \left (7 t \right ) \\ \bullet & {} & \textrm {2nd solution of the ODE}\hspace {3pt} \\ {} & {} & y_{2}\left (t \right )=\sin \left (7 t \right ) \\ \bullet & {} & \textrm {General solution of the ODE}\hspace {3pt} \\ {} & {} & y=c_{1} y_{1}\left (t \right )+c_{2} y_{2}\left (t \right ) \\ \bullet & {} & \textrm {Substitute in solutions}\hspace {3pt} \\ {} & {} & y=\cos \left (7 t \right ) c_{1} +c_{2} \sin \left (7 t \right ) \end {array} \]
Maple trace
`Methods for second order ODEs: --- Trying classification methods --- trying a quadrature checking if the LODE has constant coefficients <- constant coefficients successful`
✓ Solution by Maple
Time used: 0.0 (sec). Leaf size: 17
dsolve([diff(y(t),t$2)+49*y(t)=0,sin(7*t)],singsol=all)
\[ y \left (t \right ) = c_{1} \sin \left (7 t \right )+c_{2} \cos \left (7 t \right ) \]
✓ Solution by Mathematica
Time used: 0.014 (sec). Leaf size: 20
DSolve[y''[t]+49*y[t]==0,y[t],t,IncludeSingularSolutions -> True]
\[ y(t)\to c_1 \cos (7 t)+c_2 \sin (7 t) \]