3.10 problem 10

Internal problem ID [12937]
Internal file name [OUTPUT/11590_Tuesday_November_07_2023_11_27_34_PM_29147239/index.tex]

Book: DIFFERENTIAL EQUATIONS by Paul Blanchard, Robert L. Devaney, Glen R. Hall. 4th edition. Brooks/Cole. Boston, USA. 2012
Section: Chapter 1. First-Order Differential Equations. Exercises section 1.4 page 61
Problem number: 10.
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.

\[ \boxed {y^{\prime }-2 y^{3}=t^{2}} \] With initial conditions \begin {align*} \left [y \left (0\right ) = -{\frac {1}{2}}\right ] \end {align*}

3.10.1 Existence and uniqueness analysis

This is non linear first order ODE. In canonical form it is written as \begin {align*} y^{\prime } &= f(t,y)\\ &= 2 y^{3}+t^{2} \end {align*}

The \(t\) domain of \(f(t,y)\) when \(y=-{\frac {1}{2}}\) is \[ \{-\infty

The \(y\) domain of \(\frac {\partial f}{\partial y}\) when \(t=0\) is \[ \{-\infty

3.10.2 Solving as abelFirstKind ode

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 }&=2 y^{3}+t^{2}\tag {1} \end {align*}

Therefore \begin {align*} f_0(t) &= t^{2}\\ f_1(t) &= 0\\ f_2(t) &= 0\\ f_3(t) &= 2 \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 {4}{27 t^{7}} \end {align*}

Since the Abel invariant depends on \(t\) then unable to solve this ode at this time.

3.10.3 Maple step by step solution

\[ \begin {array}{lll} & {} & \textrm {Let's solve}\hspace {3pt} \\ {} & {} & \left [y^{\prime }-2 y^{3}=t^{2}, y \left (0\right )=-\frac {1}{2}\right ] \\ \bullet & {} & \textrm {Highest derivative means the order of the ODE is}\hspace {3pt} 1 \\ {} & {} & y^{\prime } \\ \bullet & {} & \textrm {Solve for the highest derivative}\hspace {3pt} \\ {} & {} & y^{\prime }=2 y^{3}+t^{2} \\ \bullet & {} & \textrm {Use initial condition}\hspace {3pt} y \left (0\right )=-\frac {1}{2} \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solve for}\hspace {3pt} 0 \\ {} & {} & 0=0 \\ \bullet & {} & \textrm {Substitute}\hspace {3pt} 0=0\hspace {3pt}\textrm {into general solution and simplify}\hspace {3pt} \\ {} & {} & 0 \\ \bullet & {} & \textrm {Solution to the IVP}\hspace {3pt} \\ {} & {} & 0 \end {array} \]

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)=2*y(t)^3+t^2,y(0) = -1/2],y(t), singsol=all)
 

\[ \text {No solution found} \]

✗ Solution by Mathematica

Time used: 0.0 (sec). Leaf size: 0

DSolve[{y'[t]==2*y[t]^3+t^2,{y[0]==-1/2}},y[t],t,IncludeSingularSolutions -> True]
 

Not solved