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Differential Equations/Formation of differential equations

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From Differential Equations

In the introduction we briefly discussed forming a differential equation from a function, specifically,

y = a \sin x + b \cos x, \,

and we finished seeing that

\frac{d^2y}{dx^2} = -y

and this is obviously a differential equation of the second order, because the highest order derivative involved is the second derivative. In further lessons, we can apply this to understanding the concept of the solution to a differential equation.

In general, we can obtain an ordinary differential equation with any relationship between x and y with n arbitrary constants,

f(x,y,C1,C2,...,Cn) = 0

We can take the derivatives of this in respect to x:

{\partial f \over \partial x} + {\partial f \over \partial y}y' = 0
{\partial^2 f \over \partial x^2} + 2{\partial^2 f \over \partial x \partial y}y' + {\partial^2 f \over \partial y^2}y'^2 + {\partial f \over \partial y}y'' = 0

and so on.

If the partial derivative of f in respect to y is not 0, then each of these equations are distinct and we can continue this process until we have at least n+1 equations, with which we can eliminate the n constants.

Examples
Problems
Solutions

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