# Differential Equation-Separable Equations

• neshepard
In summary, the conversation involved finding the solution of a differential equation with a given initial coordinate. The attempted solution involved a sign error and the need to substitute 0 for both x and y in order to satisfy the initial condition. The conversation also clarified that in a different problem with a different initial condition, the variables x and y were replaced by a new function u, and the same substitution process would apply.
neshepard

## Homework Statement

Find the solution of the diff eq that satisfies the given initial coordinate

## Homework Equations

xcosx=(2y + e^(3y)) y' , y(0)=0

## The Attempt at a Solution

So I have the family of solutions:
xsinx-cosx=y^2 + e^(3y)/3 + c

and I know to put 0 in for the x's, but the solution is wrong, and it appears like I need to put 0 into the y's (yes I did say y's) but not sure why.

neshepard said:

## Homework Statement

Find the solution of the diff eq that satisfies the given initial coordinate

## Homework Equations

xcosx=(2y + e^(3y)) y' , y(0)=0

## The Attempt at a Solution

So I have the family of solutions:
xsinx-cosx=y^2 + e^(3y)/3 + c

and I know to put 0 in for the x's, but the solution is wrong, and it appears like I need to put 0 into the y's (yes I did say y's) but not sure why.
Yes, substitute 0 for x and 0 for y. That's what y(0) = 0 means.

You have a sign error in your solution. It should be xsinx + cosx = y^2 + e^(3y) + C.

Thanks for the reply. And just to clarify, I have another problem where the initial condition is u(0)=-5 so I put in -5 for my x's and 0 for my y's?

neshepard said:
Thanks for the reply. And just to clarify, I have another problem where the initial condition is u(0)=-5 so I put in -5 for my x's and 0 for my y's?

If x is the dependent variable, that is, you have u(x), then you need to set x=0 and u=-5. You don't have x and y anymore, one of them has been replaced by a new function u in this other problem.

Nice...thanks.

## 1. What are differential equations and how are they different from regular equations?

Differential equations are mathematical equations that describe the relationship between a function and its derivatives. They are different from regular equations because they involve derivatives, which represent rates of change, rather than just variables and constants.

## 2. What is a separable equation and how is it solved?

A separable equation is a type of differential equation where the variables can be separated and solved independently. This means that the dependent variable can be written as a product of two functions, each of which only depends on one of the independent variables. To solve a separable equation, we use the method of separation of variables, where we move all the terms containing the dependent variable to one side of the equation and all the terms containing the independent variable to the other side. Then, we integrate both sides to obtain the general solution.

## 3. What is the general solution of a separable equation?

The general solution of a separable equation is the solution that includes all possible solutions. It is obtained by integrating both sides of the equation after separating the variables. The general solution will include an arbitrary constant, which can be determined by applying initial or boundary conditions.

## 4. Can all differential equations be solved by separation of variables?

No, not all differential equations can be solved by separation of variables. This method only works for separable equations, which are a specific type of differential equation. Other methods, such as substitution and variation of parameters, may be needed to solve other types of differential equations.

## 5. How are separable equations used in real-world applications?

Separable equations are used in many areas of science, engineering, and economics to model real-world phenomena. For example, they can be used to model population growth, radioactive decay, and chemical reactions. By solving these equations, we can make predictions and better understand the behavior of these systems.

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