Internal problem ID [14515]
Internal file name [OUTPUT/14196_Monday_March_25_2024_09_51_01_PM_77752989/index.tex]
Book: INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton.
Fourth edition 2014. ElScAe. 2014
Section: Chapter 4. Higher Order Equations. Exercises 4.2, page 147
Problem number: 42 (a).
ODE order: 2.
ODE degree: 2.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[[_2nd_order, _missing_x]]
Unable to solve or complete the solution.
\[ \boxed {{y^{\prime \prime }}^{2}-5 y^{\prime } y^{\prime \prime }+4 y^{2}=0} \] Does not support ODE with \({y^{\prime \prime }}^{n}\) where \(n\neq 1\) unless \(4 y\) is missing which is not the case here.
Maple trace
`Methods for second order ODEs: *** Sublevel 2 *** Methods for second order ODEs: Successful isolation of d^2y/dt^2: 2 solutions were found. Trying to solve each resulting ODE. *** Sublevel 3 *** Methods for second order ODEs: --- Trying classification methods --- trying 2nd order Liouville trying 2nd order WeierstrassP trying 2nd order JacobiSN differential order: 2; trying a linearization to 3rd order trying 2nd order ODE linearizable_by_differentiation trying 2nd order, 2 integrating factors of the form mu(x,y) trying differential order: 2; missing variables `, `-> Computing symmetries using: way = 3 Try integration with the canonical coordinates of the symmetry [0, y] -> Calling odsolve with the ODE`, diff(_b(_a), _a) = -_b(_a)^2+(5/2)*_b(_a)-(1/2)*(25*_b(_a)^2-16)^(1/2), _b(_a), explicit, HI symmetry methods on request `, `1st order, trying reduction of order with given symmetries:`[1, 0]
✓ Solution by Maple
Time used: 0.172 (sec). Leaf size: 85
dsolve(diff(y(t),t$2)^2-5*diff(y(t),t$2)*diff(y(t),t)+4*y(t)^2=0,y(t), singsol=all)
\begin{align*} y \left (t \right ) &= 0 \\ y \left (t \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (t +2 \left (\int _{}^{\textit {\_Z}}\frac {1}{2 \textit {\_f}^{2}+\sqrt {25 \textit {\_f}^{2}-16}-5 \textit {\_f}}d \textit {\_f} \right )+c_{1} \right )d t +c_{2}} \\ y \left (t \right ) &= {\mathrm e}^{\int \operatorname {RootOf}\left (t -2 \left (\int _{}^{\textit {\_Z}}-\frac {1}{2 \textit {\_f}^{2}-\sqrt {25 \textit {\_f}^{2}-16}-5 \textit {\_f}}d \textit {\_f} \right )+c_{1} \right )d t +c_{2}} \\ \end{align*}
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y''[t]^2-5*y''[t]*y'[t]+4*y[t]^2==0,y[t],t,IncludeSingularSolutions -> True]
Not solved