Internal problem ID [4914]
Internal file name [OUTPUT/4407_Sunday_June_05_2022_01_16_02_PM_26610584/index.tex]
Book: Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston.
Pearson 2018.
Section: Chapter 2, First order differential equations. Section 2.2, Separable Equations. Exercises.
page 46
Problem number: 3.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "unknown"
Maple gives the following as the ode type
[`y=_G(x,y')`]
Unable to solve or complete the solution.
\[ \boxed {s^{\prime }-t \ln \left (s^{2 t}\right )=8 t^{2}} \] Unable to determine ODE type.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & s^{\prime }-t \ln \left (s^{2 t}\right )=8 t^{2} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & s^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & s^{\prime }=t \ln \left (s^{2 t}\right )+8 t^{2} \end {array} \]
Maple trace
`Methods for first order ODEs: --- Trying classification methods --- trying a quadrature trying 1st order linear trying Bernoulli trying separable trying inverse linear trying homogeneous types: trying Chini differential order: 1; looking for linear symmetries trying exact Looking for potential symmetries trying inverse_Riccati trying an equivalence to an Abel ODE differential order: 1; trying a linearization to 2nd order --- trying a change of variables {x -> y(x), y(x) -> x} differential order: 1; trying a linearization to 2nd order trying 1st order ODE linearizable_by_differentiation --- Trying Lie symmetry methods, 1st order --- `, `-> Computing symmetries using: way = 3 `, `-> Computing symmetries using: way = 5 trying symmetry patterns for 1st order ODEs -> trying a symmetry pattern of the form [F(x)*G(y), 0] -> trying a symmetry pattern of the form [0, F(x)*G(y)] -> trying symmetry patterns of the forms [F(x),G(y)] and [G(y),F(x)] -> trying a symmetry pattern of the form [F(x),G(x)] -> trying a symmetry pattern of the form [F(y),G(y)] -> trying a symmetry pattern of the form [F(x)+G(y), 0] -> trying a symmetry pattern of the form [0, F(x)+G(y)] -> trying a symmetry pattern of the form [F(x),G(x)*y+H(x)] -> trying a symmetry pattern of conformal type`
✗ Solution by Maple
dsolve(diff(s(t),t)=t*ln(s(t)^(2*t))+8*t^2,s(t), singsol=all)
\[ \text {No solution found} \]
✓ Solution by Mathematica
Time used: 0.28 (sec). Leaf size: 34
DSolve[s'[t]==t*Log[s[t]^(2*t)]+8*t^2,s[t],t,IncludeSingularSolutions -> True]
\begin{align*} s(t)\to \text {InverseFunction}\left [\frac {\operatorname {ExpIntegralEi}(\log (\text {$\#$1})+4)}{e^4}\&\right ]\left [\frac {2 t^3}{3}+c_1\right ] \\ s(t)\to \frac {1}{e^4} \\ \end{align*}