Find the unique solution to the IVP

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In summary, the given IVP has a unique solution, which can be found by testing the trivial solution $y(t)=0$. No further steps are necessary to find the solution.
  • #1
shamieh
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Find the unique solution to the IVP

$t^3y'' + e^ty' + t^4y = 0$ $y(1) = 0$ , $y'(1) = 0$

Should I start out by dividing through by $t^4$

to get

$\frac{1}{t} y" + \frac{e^t}{t^4}y' + y = 0$
 
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  • #2
I think in this case, you should look for a trivial solution to the given ODE. :D
 
  • #3
what do you mean? should I plug y(1) = 0
 
  • #4
Consider the function:

\(\displaystyle y(t)=0\)

Does it satisfy all of the given requirements?
 
  • #5
Yes?
 
  • #6
MarkFL said:
Consider the function:

\(\displaystyle y(t)=0\)

Does it satisfy all of the given requirements?

Oh wait.. I need to test this using the exact solution method don't i?

(testing whether it is exact or not)
 
  • #7
shamieh said:
Oh wait.. I need to test this using the exact solution method don't i?

(testing whether it is exact or not)

That's for first order equations...to be honest, I would not know offhand how to find the general solution to the ODE associated with the IVP here, but I simply noticed that the trivial solution $y(t)=0$ satisfies the IVP. :D
 

FAQ: Find the unique solution to the IVP

What is an IVP?

An IVP stands for initial value problem. It is a type of mathematical problem that involves finding a solution to a differential equation, given some initial conditions.

What is a unique solution?

A unique solution to an IVP means that there is only one possible solution that satisfies both the differential equation and the given initial conditions.

How do you find the unique solution to an IVP?

To find the unique solution to an IVP, you need to use a method called separation of variables. This involves separating the variables in the differential equation, integrating both sides, and then solving for the constant of integration to find the specific solution.

Why is it important to find the unique solution to an IVP?

Finding the unique solution to an IVP is important because it allows us to accurately model and predict the behavior of a system or process. It also ensures that there is only one possible outcome, eliminating any ambiguity or uncertainty.

What are some real-life applications of finding the unique solution to an IVP?

The unique solution to an IVP has many real-life applications, such as predicting population growth, modeling the spread of diseases, and analyzing chemical reactions. It is also used in fields such as physics, engineering, economics, and biology to understand and solve complex problems.

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