Introduction:
In algebraic equations, the unknown is a number. Whereas in differential equations,
the unknown is a function.
The difficult part about solving differential equations is first determining if there exists
a solution, and then finding the solution.
Classification of differential equations:
Ordinary differential equations (ODE) involve a function of a single independent
variable.
Partial differential equations (PDE) involve a function of more than one independent
variable, and partial derivatives.
The order of a differential equation is the order of the highest derivative it contains.
Linear ODE:
An ODE is linear if it can be written in the form:
n n−1
d y d y
a n ( x ) n + an−1 ( x ) + …+a0 ( x ) y =g( x) where a n ( x ) … a0 ( x ) are functions of x only.
dx d x n−1
Separable ODE:
An ODE is separable if it can be written in the form:
dy
=Y ( y ) X ( x)where Y is a function of y only, and X is a function of x only.
dx
These are solved by separating the variables, moving each to one side, and then
integrating both sides.
First order linear ODE:
These are of the form:
dy
a1 ( x ) + a ( x ) y =h( x )where a 1 ( x ) ,a 0 ( x ) , h(x) are functions of x only. If h ( x ) = 0, this
dx 0
DE is separable.
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