-b.1.3.12 .... is a solution of the DE

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Discussion Overview

The discussion revolves around verifying that the functions $y_2(t)=t^{-2}\ln t$ and $y_1(t)=t^{-2}$ are solutions to the differential equation $t^2y''+5ty'+4y=0$. The scope includes mathematical reasoning and verification of solutions to differential equations.

Discussion Character

  • Mathematical reasoning
  • Technical explanation

Main Points Raised

  • One participant suggests starting with the characteristic equation using the variable $r$.
  • Another participant calculates the derivatives of $y_1(t)$ and substitutes them into the differential equation, concluding that the equation holds true.
  • A participant expresses that the verification process is straightforward compared to solving the equation itself, implying a preference for verification over solving.
  • A later reply emphasizes that the task is merely to verify the solutions by substitution, contrasting it with the more complex process of solving the equation.

Areas of Agreement / Disagreement

Participants generally agree on the verification process for the solutions, but there is no consensus on the complexity of solving the differential equation itself, as some find it mind-numbing while others do not express a clear opinion.

Contextual Notes

Limitations include the potential complexity of solving the differential equation, which is not fully explored in the discussion. The verification process is presented without addressing any assumptions or definitions that may affect the results.

karush
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#12 hope I rewrote the problem ok

Verify that $y_2(t)=t^{-2}\ln t\quad y_1(t)=t^{-2}$ is a solution of $t^2y''+5ty'+4y=0$
think the first steop is to compose a charactistic equation using r
 

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$y_1(t) = t^{-2} \implies y'_1(t) = -2t^{-3} \implies y''_1(t) = 6t^{-4}$

$t^2 \cdot 6t^{-4} + 5t \cdot(-2t^{-3}) + 4 \cdot t^{-2} = 6t^{-2} - 10t^{-2} + 4t^{-2} = 0$

verified
 
so that is how it is done...
the examples were pretty mind numbing compared

I really think MHB should write textbooks...
 
karush said:
so that is how it is done...
the examples were pretty mind numbing compared

I really think MHB should write textbooks...
You are merely supposed to verify the solutions. So just plug them in. Solving the equation, on the other hand, can be a bit mind numbing until you get used to it.

-Dan
 

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