# Separable Equation with Condition?

• silicon_hobo
In summary, the conversation discusses solving a separable equation with the additional condition of y(2)=2. The process involves factoring and integrating both sides, with the initial condition used to solve for an arbitrary constant.
silicon_hobo
[SOLVED] Separable Equation with Condition?

## Homework Statement

Solve.
http://www.mcp-server.com/~lush/shillmud/int3.71.JPG

## The Attempt at a Solution

I'm not sure how to separate this. Also, since the directions consist of only one word, I'm not sure if y(2)=2 is some kind of hint or an additional condition to be fulfilled. I'm wondering where to go after factoring t out of the top. Thanks.
http://www.mcp-server.com/~lush/shillmud/int3.72.JPG

Well after factoring... you separate (hence separable equations!)

$$\frac{dy}{y+3} = \frac{t}{t^2+1}dt$$

Then integrate both sides.
y(2) = 2 is an initial condition, not a hint... After solving for y(t) you will have one arbitrary constant, which can be solved for using this condition.

Wow, that seems almost too easy. Thanks.

## 1. What is a separable equation with condition?

A separable equation with condition is a type of differential equation where the dependent variable and independent variable can be separated into separate functions, with the addition of a condition that must also be satisfied.

## 2. What is the purpose of adding a condition to a separable equation?

The condition in a separable equation ensures that the solution to the equation satisfies a specific requirement or boundary. This can be helpful in solving real-world problems or providing more accurate solutions.

## 3. How do you know if an equation is separable with condition?

An equation is separable with condition if it can be written in the form dy/dx = f(x)g(y), where f(x) and g(y) are functions of x and y respectively, and there is an additional condition that must be satisfied.

## 4. Can you give an example of a separable equation with condition?

One example of a separable equation with condition is the equation dy/dx = x/y, where the condition is that y cannot be equal to 0. This equation can be separated into dy/y = xdx, and then solved using integration techniques.

## 5. What are the advantages of solving a separable equation with condition?

The main advantage of solving a separable equation with condition is that it allows for more precise solutions to problems. It also allows for the incorporation of real-world constraints or boundary conditions, making the solution more applicable to practical situations.

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