A first order differential equation is of the form:
The general general solution is given by
where
is called the integrating factor.
is homogeneous if the function f(x,y) is homogeneous, that is
By substitution, we consider the new function
The new differential equation satisfied by z is
which is a separable equation. The solutions are the constant ones f(1,z) - z =0 and the non-constant ones given by
Do not forget to go back to the old function y = xz.
is exact if
The condition of exactness insures the existence of a function F(x,y) such that
All the solutions are given by the implicit equation
Write down the characteristic equation
where
that is
The general solution is given by
where is a particular solution and is the general solution of the associated homogeneous equation
In order to find two major techniques were developed.
where a, b, and c are constant and
where is a polynomial function with degree n. In this case, we have
where
The constants and have to be determined. The power
s is equal to 0 if is not a root of the
characteristic equation. If is a simple root, then
s=1 and s=2 if it is a double root.
Remark. If the nonhomogeneous term g(x)
satisfies the following
where are of the forms cited above, then we split the original equation into N equations
then find a particular solution . A particular solution to the original equation is given by
Note that this method works regardless if the coefficients are constant or not. a particular solution as
where and are functions. From this, the method got its name.
The functions and are solutions to the system:
which implies
Therefore, we have
where b and c are constant numbers. By substitution, set
then the new equation satisfied by y(t) is
which is a second order differential equation with constant coefficients.
where and .
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