# Separable D.E., Nonlinear in terms of y(x) after integration

• EtherealMonkey
In summary, a separable differential equation is one in which the variables can be separated and each side of the equation can be integrated separately. After integration, the resulting equation is nonlinear in terms of y(x) if it cannot be written as a linear combination of y(x) and its derivatives. Separation of variables can be used to solve a separable differential equation by isolating the dependent variable and integrating both sides of the equation. The purpose of integrating is to find the general solution, which can then be used to find specific solutions for different initial conditions. However, there are limitations to using separation of variables as it may not always be possible to separate the variables and the resulting equation may not be solvable in terms of elementary functions.
EtherealMonkey
So, this is where I am stuck:

$$ln\left(y\right)+y^{2} = \sin{x}+c_{0}$$

I am confrused...

Do you think that

$$y(x) =\pm\sqrt{2LambertW(2exp(2sin(x)+C)}/2$$

is better?

Last edited by a moderator:
It's easier to express <x> in terms of <y>, wouldn't you say ?

EtherealMonkey, are you required to solve for y? That isn't always necessary or possible.

hmm, but is it possible to make an explicit form in terms of x?

kosovtsov just posted it...

waw, lambert, i need to study that thing

## 1. What is a separable differential equation?

A separable differential equation is one in which the variables can be separated and each side of the equation can be integrated separately.

## 2. How is a differential equation nonlinear in terms of y(x) after integration?

A differential equation is nonlinear in terms of y(x) after integration if the resulting equation cannot be written as a linear combination of y(x) and its derivatives.

## 3. Can a separable differential equation be solved using separation of variables?

Yes, a separable differential equation can be solved using separation of variables, which involves isolating the dependent variable and integrating both sides of the equation.

## 4. What is the purpose of integrating a separable differential equation?

Integrating a separable differential equation allows us to find the general solution to the equation, which can then be used to find specific solutions for different initial conditions.

## 5. Are there any limitations to using separation of variables to solve a separable differential equation?

Yes, there are some limitations to using separation of variables, as it may not be possible to separate the variables in certain cases. Additionally, the resulting equation may not always be solvable in terms of elementary functions.

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