Find the Particular Solution for ds/dt = 14t^2+3t-3 with s=124 at t=0

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Discussion Overview

The discussion revolves around finding the particular solution to the differential equation ds/dt = 14t^2 + 3t - 3, given the initial condition s = 124 when t = 0. The scope includes mathematical reasoning and integration techniques.

Discussion Character

  • Mathematical reasoning
  • Homework-related

Main Points Raised

  • One participant requests guidance on how to start solving the problem, expressing uncertainty about the title of the post.
  • Another participant suggests using integration to find the solution, providing a hint that involves integrating the right-hand side of the equation.
  • A different approach is proposed, where definite integrals are set up with the initial conditions as limits, suggesting a method to find the particular solution.
  • Another participant reiterates the need to integrate the function and provides the indefinite integral, indicating the necessity to evaluate the constant using the initial condition.
  • This participant concludes with a proposed form of the solution, including the constant determined from the initial condition.

Areas of Agreement / Disagreement

Participants generally agree on the need to integrate the function to find the solution and on the approach of using the initial condition to evaluate the constant. However, there are multiple methods suggested for setting up the integration, and no consensus is reached on the preferred method.

Contextual Notes

Some participants express uncertainty about the integration process and the implications of the initial condition, indicating that there may be assumptions or steps that are not fully articulated.

hallie
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Find the particular solution determined by the given condition:

ds/dt = 14t^2+3t -3; s=124 when t = 0

Could someone please point me in the right direction to start this problem? I'm not even sure if I titled this post correctly. :P

Thank you!
 
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hallie said:
Find the particular solution determined by the given condition:

ds/dt = 14t^2+3t -3; s=124 when t = 0

Could someone please point me in the right direction to start this problem? I'm not even sure if I titled this post correctly. :P

Thank you!
Hint: [math]\int \frac{ds}{dt}~dt = \int ( 14t^2+3t -3)~dt[/math]

And the LHS is equal to s(t).

-Dan
 
You could also set it up using definite integrals by switching the dummy variables of integration and using the given boundaries as the limits:

$$\int_{124}^{s(t)} \,du=\int_0^t 14v^2+3v -3\,dv$$
 
hallie said:
Find the particular solution determined by the given condition:

\frac{ds}{dt} \: =\: 14t^2+3t -3;\;\; s=124 \text{ when }t = 0.

Could someone please point me in the right direction to start this problem?
It seems that you are new to this type of problem.
Do you understand the given statements?

We are given the derivative of the s-function
and we are asked to find the s-function.

You should know that we will integrate the given function,

\int(14t^2 + 3t - 3)\,dt \;=\;\tfrac{14}{3}t^3 + \tfrac{3}{2}t^2 - 3t + C

We must evaluate that constant, using the given initial condition.
\text{When }t = 0, s = 124.

So we have: \tfrac{14}{3}(0^3) + \tfrac{3}{2}(0^2) - 3(0) \:=\:124 \quad\Rightarrow\quad C = 124

Answer: .s(t) \;=\;\tfrac{14}{3}t^3 + \tfrac{3}{2}t^2 - 3t + 124
 

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