# Differential Equation/Intial Value Problem

• tracedinair
In summary, a differential equation is a mathematical equation that describes how a quantity changes over time, relating the rate of change to the value and derivatives of a variable. An initial value problem is a type of differential equation that involves finding a solution for a function given an initial value or condition. Differential equations have wide-ranging applications in fields such as physics, engineering, economics, and biology. Not all differential equations can be solved analytically, and in these cases, numerical methods may be used. The difference between ordinary and partial differential equations lies in the number of independent variables and the derivatives being taken.
tracedinair

## Homework Statement

Solve the following initial value problem,

y' = (x-y)/(x+y), y(1) = 1

## The Attempt at a Solution

dy/dx = (x-y)/(x+y)

dy/dx * (x+y) = x-y

(x+y)dy = (x-y)dx

∫(x+y)dy = ∫(x-y)dx

xy + y^(2)/2 = x^(2)/2 - xy + C

2xy + y^2 = x^2 - 2xy + C

y^2 + 4xy - x^2 = C

Sub in x=1, y=1

1 + 4 - 1 = C = 4

Soln: 2xy + y^2 = x^2 - 2xy + 4

N/m, I just realized this is a homogeneous diffeq. Sorry.

## What is a differential equation?

A differential equation is a mathematical equation that describes how a quantity changes over time. It relates the rate of change of a variable to the value of the variable and its derivatives.

## What is an initial value problem?

An initial value problem is a type of differential equation that involves finding the solution for a function given an initial value or condition. It typically includes a specific starting point or condition that the function must satisfy.

## What are the applications of differential equations?

Differential equations have wide-ranging applications in various fields such as physics, engineering, economics, biology, and more. They are used to model and predict the behavior of complex systems and phenomena.

## Can all differential equations be solved analytically?

No, not all differential equations can be solved analytically. Some equations may have no explicit solution or may require advanced mathematical techniques to solve. In these cases, numerical methods may be used to approximate a solution.

## What is the difference between ordinary and partial differential equations?

Ordinary differential equations involve a single independent variable, while partial differential equations involve multiple independent variables. Additionally, the derivatives in ordinary differential equations are with respect to the single variable, while in partial differential equations they are with respect to multiple variables.

• Calculus and Beyond Homework Help
Replies
3
Views
464
• Calculus and Beyond Homework Help
Replies
25
Views
730
• Calculus and Beyond Homework Help
Replies
4
Views
308
• Calculus and Beyond Homework Help
Replies
10
Views
734
• Calculus and Beyond Homework Help
Replies
6
Views
969
• Calculus and Beyond Homework Help
Replies
2
Views
730
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
8
Views
922
• Calculus and Beyond Homework Help
Replies
14
Views
693
• Calculus and Beyond Homework Help
Replies
21
Views
1K