Find the unique solution to the IVP

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Discussion Overview

The discussion revolves around finding the unique solution to the initial value problem (IVP) defined by the differential equation $t^3y'' + e^ty' + t^4y = 0$ with initial conditions $y(1) = 0$ and $y'(1) = 0$. Participants explore potential solutions and methods for solving the equation.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested, Homework-related

Main Points Raised

  • One participant suggests starting by dividing through by $t^4$ to reformulate the equation.
  • Another participant proposes looking for a trivial solution to the ODE.
  • There is a question about whether plugging in the initial condition $y(1) = 0$ is necessary for finding a solution.
  • A participant considers the function $y(t) = 0$ and questions if it meets all the requirements of the IVP.
  • One participant expresses uncertainty about how to find the general solution to the ODE associated with the IVP but notes that the trivial solution satisfies the IVP.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the method for solving the IVP, and multiple approaches are discussed without resolution.

Contextual Notes

There is uncertainty regarding the appropriate method for solving the ODE, with some participants suggesting different approaches and questioning the necessity of testing the trivial solution against the initial conditions.

Who May Find This Useful

Students and individuals interested in differential equations, particularly those studying initial value problems and solution methods.

shamieh
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Find the unique solution to the IVP

$t^3y'' + e^ty' + t^4y = 0$ $y(1) = 0$ , $y'(1) = 0$

Should I start out by dividing through by $t^4$

to get

$\frac{1}{t} y" + \frac{e^t}{t^4}y' + y = 0$
 
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I think in this case, you should look for a trivial solution to the given ODE. :D
 
what do you mean? should I plug y(1) = 0
 
Consider the function:

$$y(t)=0$$

Does it satisfy all of the given requirements?
 
Yes?
 
MarkFL said:
Consider the function:

$$y(t)=0$$

Does it satisfy all of the given requirements?

Oh wait.. I need to test this using the exact solution method don't i?

(testing whether it is exact or not)
 
shamieh said:
Oh wait.. I need to test this using the exact solution method don't i?

(testing whether it is exact or not)

That's for first order equations...to be honest, I would not know offhand how to find the general solution to the ODE associated with the IVP here, but I simply noticed that the trivial solution $y(t)=0$ satisfies the IVP. :D
 

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