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

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The discussion focuses on 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 verification process involves calculating the derivatives of y_1(t) and substituting them into the equation, which simplifies to zero, confirming that y_1(t) is indeed a solution. Participants express that while the verification is straightforward, solving the equation can be tedious. There is a suggestion that the forum, MHB, should consider writing textbooks due to the complexity of the examples discussed. Overall, the emphasis is on the verification process rather than solving the differential equation itself.
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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