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

• MHB
• karush
In summary, the conversation discusses verifying the solutions of a given equation using a characteristic equation, with the example of $y_1(t) = t^{-2}$ and $y_2(t)=t^{-2}\ln t$. The process involves plugging in the solutions and solving the equation, which may be challenging. The suggestion is made that MHB should write textbooks to simplify the process.
karush
Gold Member
MHB
####
#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

## What is a solution of the DE?

A solution of the DE, or differential equation, is a mathematical function that satisfies the given equation when substituted into it. It is essentially the "answer" to the equation.

## How do you know if a function is a solution of the DE?

To determine if a function is a solution of the DE, you can substitute it into the equation and see if it satisfies the equation. If it does, then it is a solution. Additionally, you can also use methods such as separation of variables or variation of parameters to solve the DE and verify the solution.

## Can there be multiple solutions to a single DE?

Yes, it is possible for a single DE to have multiple solutions. This is because there are often many different functions that can satisfy the given equation. However, some DEs may only have a unique solution.

## What is the importance of finding solutions to DEs?

DEs are used to model many real-world phenomena in fields such as physics, engineering, and economics. Finding solutions to DEs allows us to make predictions and understand the behavior of these systems. They also play a crucial role in developing new technologies and solving complex problems.

## Are there any limitations to finding solutions to DEs?

Yes, there are some limitations to finding solutions to DEs. Some equations may not have closed-form solutions, meaning they cannot be expressed in terms of familiar mathematical functions. In these cases, numerical methods or approximations may be used to find solutions. Additionally, some DEs may have no solutions or only have solutions that are valid for a limited range of inputs.

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