Internal problem ID [1704]
Internal file name [OUTPUT/1705_Sunday_June_05_2022_02_28_11_AM_48880862/index.tex]
Book: Differential equations and their applications, 3rd ed., M. Braun
Section: Section 1.10. Page 80
Problem number: 11.
ODE order: 1.
ODE degree: 1.
The type(s) of ODE detected by this program : "abelFirstKind"
Maple gives the following as the ode type
[_Abel]
Unable to solve or complete the solution.
\[ \boxed {y^{\prime }-y^{3}={\mathrm e}^{-5 t}} \]
This is Abel first kind ODE, it has the form \[ y^{\prime }= f_0(t)+f_1(t) y +f_2(t)y^{2}+f_3(t)y^{3} \] Comparing the above to given ODE which is \begin {align*} y^{\prime }&=y^{3}+{\mathrm e}^{-5 t}\tag {1} \end {align*}
Therefore \begin {align*} f_0(t) &= {\mathrm e}^{-5 t}\\ f_1(t) &= 0\\ f_2(t) &= 0\\ f_3(t) &= 1 \end {align*}
Since \(f_2(t)=0\) then we check the Abel invariant to see if it depends on \(t\) or not. The Abel invariant is given by \begin {align*} -\frac {f_{1}^{3}}{f_{0}^{2} f_{3}} \end {align*}
Which when evaluating gives \begin {align*} -\frac {125 \,{\mathrm e}^{10 t}}{27} \end {align*}
Since the Abel invariant depends on \(t\) then unable to solve this ode at this time.
\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & y^{\prime }-y^{3}={\mathrm e}^{-5 t} \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=y^{3}+{\mathrm e}^{-5 t} \end {array} \]
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 trying Abel Looking for potential symmetries Looking for potential symmetries 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 = 4 `, `-> Computing symmetries using: way = 2 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(y(t),t)= y(t)^3+exp(-5*t),y(t), singsol=all)
\[ \text {No solution found} \]
✗ Solution by Mathematica
Time used: 0.0 (sec). Leaf size: 0
DSolve[y'[t]== y[t]^3+Exp[-5*t],y[t],t,IncludeSingularSolutions -> True]
Not solved