What's the General Solution to This IVP?

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Discussion Overview

The discussion centers around the initial value problem (IVP) defined by the differential equation \(\frac{dy}{dt} = y^3 + t^2\) with the initial condition \(y(0) = 0\). Participants are exploring the nature of the solution and whether it can be expressed in terms of known functions.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant inquires about the general solution to the IVP, noting their teacher's claim that it cannot be expressed in terms of commonly known functions.
  • Another participant suggests that the equation is in Bernoulli form and references a video for solving it, although this claim is later contested.
  • A different participant argues that the equation does not fit the Bernoulli form, stating it does not match the required structure for such equations.
  • Another participant expresses uncertainty about how to solve the equation and mentions that WolframAlpha provides a simple answer of \(x=1\) for \(y=0\), questioning how that result is derived.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the equation or its solution. There are competing views regarding whether it is a Bernoulli equation, and uncertainty remains about the correct approach to solving it.

Contextual Notes

There is a lack of clarity regarding the classification of the differential equation and the assumptions underlying the proposed methods of solution. The discussion reflects differing interpretations of the equation's form and the implications for finding a solution.

logan3
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[itex]\frac {dy}{dt} = y^3 + t^2, y(0) = 0[/itex]

My teacher said this IVP couldn't be expressed in terms of functions we commonly know. I was wondering what the general solution is?

Thank-you
 
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logan3 said:
[itex]\frac {dy}{dt} = y^3 + t^2, y(0) = 0[/itex]

My teacher said this IVP couldn't be expressed in terms of functions we commonly know. I was wondering what the general solution is?

Thank-you
For solving this
It is a bernoulli form
Watch this video and you will get it how to solve
 
##\frac {dy}{dt} = y^3 + t^2, y(0) = 0## is not of the form ##y' + f(t)y = g(t)y^n## so it is not the Bernoulli equation.
 
Yes it's not Bernoulli form, sorry. I not know how to solve it, but Wolframalpha was showing the simple answer x=1 for y=0.
I don't know how it solved that.
 

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