Second-Order Differential Equation

Second-Order Differential Equation

A second-order differential equation is called linear if it can be written in the form:
?(P(x) (d^2 y)/?dx?^2 +Q(x) dy/dx+R(x)y=G(x) )…(1)
Where the coefficient functions P, Q, R, and G are continuous functions. Otherwise, it is callednonlinear.

When G(x)=0, for all x, such second-order lineardifferential equation is called homogeneous and has the following form:
?(P(x) (d^2 y)/?dx?^2 +Q(x) dy/dx+R(x)y=0) …(2)
IfG(x)?0 for some x, it is callednonhomogeneous.

Two basic facts enable us to solve homogeneous linear equations are stated in the following theorems:

Theorem 1 If y_1 (x) and y_2 (x) are both solutions of the linear homogeneous Equation (2) and (c_1and c_1) are any constants, then the function:
?(y(x)=?c_1 y?_1 (x)+?c_2 y?_2 (x))
is also a solution of Equation (2).

Theorem 2 If y_1 (x) and y_2 (x) are linearly independent solutions of Equation (2)on an interval, and P(x) is never 0, then the general solution is given by:
?(y(x)=?c_1 y?_1 (x)+?c_2 y?_2 (x))
wherec_1 and c_1 are arbitrary constants.

To summarize; the first of these theorems says that if we know two solutions y_1 and y_2 of such an equation, then the linear combination y=?c_1 y?_1+?c_2 y?_2 is a solution.
The other says that the general solution is a linear combination of two linearly independent solutions y_1 andy_2. This means that neither y_1nor y_2 is a constant multiple of the other. This is very useful because if we know two particular linearly independent solutions, then we know every solution.