# Differential Equation Help: As t approaches 0, y approaches

• TheCarl
In summary, the conversation involves discussing a solution to an initial value problem and finding the critical value a0. The solution for y is given as -cos(t)/(t^2) + (a*pi^2)/(4t^2), and the critical value a0 is found to be 4/pi^2. The behavior of the solution corresponding to the initial value a0 is being discussed, with the question of what y approaches as t approaches 0. The suggested approach is to use a Taylor expansion or power series for cos(t) in the vicinity of 0.
TheCarl

## Homework Statement

http://edugen.wileyplus.com/edugen/shared/assignment/test/session.quest1886032entrance1_N10020.mml?size=14&rnd=1360201586591

(b) Solve the initial value problem and find the critical value a0 exactly.
y = ?​
a0 = ?​
(c) Describe the behavior of the solution corresponding to the initial value a0.
y -> ? as t -> 0​

## The Attempt at a Solution

I got part b correct but I thought I'd put it in here to help speed the process for whoever can help me.

(b) y= -cos(t)/(t^2) + (a*pi^2)/(4t^2)

a0 = 4/pi^2

(c) I would think y would approach 0 as t approaches 0 but that apparently is wrong. This is where I need assistance. Any help is greatly appreciated.

Last edited by a moderator:
TheCarl said:
(b) y= -cos(t)/(t^2) + (a*pi^2)/(4t^2)
a0 = 4/pi^2
I assume a = a0, so y= (1-cos(t))t-2
Do you know an expansion for cos(t) valid in the vicinity of 0?

I apologize if I seem a bit dense but could you elaborate on your question about the expansion on cos(t)? I'm not entirely sure what you're asking.

Taylor expansion? Power series?

## 1. What is a differential equation?

A differential equation is a mathematical equation that relates a function to its derivatives. It is used to model and solve change over time in various fields such as physics, engineering, and economics.

## 2. Why is the limit of y approaching 0 important in differential equations?

The limit of y approaching 0 is important because it helps determine the behavior of the function as time (t) approaches 0. It is a key factor in solving differential equations and understanding the solutions.

## 3. What is the significance of t approaching 0 in differential equations?

As t approaches 0, it represents the initial condition or starting point of the function. It is also used to determine the behavior of the function as time progresses and to find the solution to the differential equation.

## 4. How do you solve a differential equation with the limit of y approaching 0?

To solve a differential equation with the limit of y approaching 0, you need to use techniques such as separation of variables, integrating factors, or substitution. You also need to use the initial condition (t = 0, y = ?) to find the specific solution.

## 5. Can differential equations be applied in real-life situations?

Yes, differential equations are widely used in various fields to model and solve problems involving change over time. They can be used to predict population growth, analyze electrical circuits, and understand the behavior of physical systems, among others.

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