# Differential Equations: Find the Solutions and Limiting Factos

• Northbysouth
Normally the solution would involve a constant of integration, but since you have an initial condition for y(1), you can find the value of that constant. In summary, the conversation discusses determining the solution and limiting factor for a given differential equation, with an initial condition of y(1) = -2. The process involves isolating y' and finding u(t) through integration, followed by solving for the constant of integration based on the given initial condition. There is some confusion about the term "limiting factor" and its significance in the problem.
Northbysouth

## Homework Statement

Determine the solution and limiting factor

(t+1)y' + y = 6

y(1) = -2

## The Attempt at a Solution

So I started off by isolating y'

y' + y/(t+1) = 6(t+1)

Then I found u(t)

u(t) = e∫ t+1 dt = et2/2 +t

y' =(et2/2 +t) = y(et2/2 +t) = 6(et2/2 +t)/(t+1)

The difficulty I'm having is integrating 6(et2/2 +t)/(t+1)

u-substitution doesn't help. Suggestions are appreciated

Northbysouth said:

## Homework Statement

Determine the solution and limiting factor

(t+1)y' + y = 6

y(1) = -2

## The Attempt at a Solution

So I started off by isolating y'

y' + y/(t+1) = 6(t+1)

Then I found u(t)

u(t) = e∫ t+1 dt

That should be ##u(t) = e^{\int \frac 1 {t+1}}\, dt##.

Ahh, thank you.

Can you explain to me what is meant by the limiting factor? Is it the largest range of t values for the given information of y(1) = -2?

Northbysouth said:
Ahh, thank you.

Can you explain to me what is meant by the limiting factor? Is it the largest range of t values for the given information of y(1) = -2?

I am not familiar with the term "limiting factor". Perhaps it refers to disallowed values of t in your answer.

## 1. What are differential equations?

Differential equations are mathematical equations that describe how a quantity changes over time, usually in terms of its rate of change. They are used to model various natural phenomena and processes in physics, engineering, and other sciences.

## 2. How do you find the solutions to differential equations?

The method for finding solutions to differential equations depends on the type of equation. Some common methods include separation of variables, integrating factors, and using series solutions. In some cases, the solutions can also be approximated numerically using computer software.

## 3. What are initial value problems in differential equations?

Initial value problems are a type of differential equation that involve finding a solution that satisfies both the equation and an initial condition, typically in the form of a given value for the dependent variable at a specific point in time.

## 4. How do you determine the limiting factors in differential equations?

Limiting factors in differential equations refer to the conditions or parameters that affect the behavior or stability of the solutions. They can be determined by analyzing the equation and its solutions, and may include factors such as boundary conditions, external forces, or physical constraints.

## 5. What are some real-world applications of differential equations?

Differential equations have numerous practical applications in various fields, such as predicting population growth, modeling chemical reactions, analyzing electrical circuits, and studying climate change. They are also widely used in engineering for designing and optimizing systems and processes.

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