# Diff.Eq. Seperation of variables.

• Jtechguy21
In summary, the conversation discusses solving a differential equation using separation of variables and then using an initial condition to determine the value of a constant. The solution involves taking integrals and solving for the constant. There is also a discussion about checking the correctness of the integration by taking the derivative.
Jtechguy21

## Homework Statement

Solve the given differential equation subject to the indicated initial condition.

(e^-y + 1)sinxdx=(1+cosx)d, y(0)=0

## Homework Equations

Basically we have to use separation of varaibles to solve before using initial value condition.

## The Attempt at a Solution

After separation of variables

Dy/(e^-y +1) = sinx dx/(1+cosx)

take the integral of both sides
∫Dy/(e^-y +1)=ln|e^-y+1|+y

∫sinx dx/(1+cosx)= -ln(1+cosx)+c

Clean it up a bit
ln|e^-y+1|+y= -ln(1+cosx)+cI have no idea what to do now with the y(0)=0
That means plus in x=0 for the equation correct?

Last edited:
You use the initial condition to determine the value of c.

when x = 0, y = 0.

1 person
SteamKing said:
You use the initial condition to determine the value of c.

when x = 0, y = 0.

How did you get y=0?

Jtechguy21 said:
How did you get y=0?

That's what $y(0) = 0$ means!

1 person
well still there is a problem if x(0) is not zero for y(0)=0...
In that case you solve c with respect to x(0) (a parameter) and input it

1 person
okay thank you.
My last question is.
can someone please check my integration
∫Dy/(e^-y +1)=ln|e^-y+1|+y

to make sure ^ is correct?

you can always try to take the derivative of the righthand side and check by yourself ;) much easier

## 1. What is Separation of Variables in Differential Equations?

Separation of Variables is a method used to solve certain types of differential equations, where the equation can be separated into two simpler equations using algebraic manipulations. This allows us to find a general solution to the original equation.

## 2. When is Separation of Variables used?

Separation of Variables is typically used when solving ordinary differential equations (ODEs) that have the form of a first-order or second-order equation with constant coefficients.

## 3. What are the steps involved in Separation of Variables?

The first step is to rearrange the equation so that all the variables on one side and the constants on the other side. Then, we divide both sides by the coefficient of the derivative term. Next, we integrate both sides with respect to the variable of interest. Finally, we use algebra to rearrange the equation and find the general solution.

## 4. What are the limitations of Separation of Variables?

Separation of Variables can only be used for certain types of differential equations, particularly first-order or second-order equations with constant coefficients. It is also limited to equations that can be rearranged and integrated using algebraic methods.

## 5. Can Separation of Variables be used for partial differential equations?

No, Separation of Variables is not applicable to partial differential equations (PDEs). It can only be used for ordinary differential equations (ODEs) that have a single independent variable.

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