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The fundamental equation that we can start with is an equation whose coefficient is not zero. It can be solved by the integration method or by the guessing method. There are three types of differential equation: Ordinary differential equation, linear differential equation and non-linear differential equation.

The method to solve the differential equation is to check the degree of the equation, if the equation has degree zero it is called an ordinary differential equation, the degree is greater than zero it is called a linear differential equation.

The method used to solve the differential equation is the separation of variables method. The fundamental idea to solving the differential equation is the separation of the variable. If you are given a differential equation in the form of y’ = f(x,y) then split the function into two parts f(x,y) = g(x,y)h(y) and the equation becomes dy/dx = g(x,y)h(y) or dy/dx = g(x)h(y). By using the initial condition, rewrite this problem in the form of 1/h(y) dy = g(x)dx and then integrate on both sides.

The other method to solve the differential equation is to guess the function. For example, if you are given a differential equation x = y f(x, y) then try to find a function f(x, y) that can be guessed. If there is no function, then you should try another guess function.

If the equation has a solution we say it is integrable and if it does not have a solution it is non-integrable. The ODE is called integrable if it has a solution that can be expressed as a series of power of e.

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