Second-Order Linear Ordinary Differential Equations

A linear second-order ODE has the form:

On any interval where S(t) is not equal to 0, the above equation can be divided by S(t) to yield

The equation is called homogeneous if f(t)=0. Otherwise, it is called nonhomogeneous.

Existence and Uniqueness

A second-order differential equation is accompanied by initial conditions or boundary conditions. Initial conditions are in the form y(t0)=y0 and y'(t0)=y'0. Boundary conditions might be of the form: y(t0)=a and y(t1)=b.

For the initial value problem, the existence and uniqueness theorem states that if p(t), q(t) and f(t) are continuous on some interval (a,b) containing t0, then there exists a unique solution y(t) to the ode in the whole interval (a,b).

Procedure for Solving Linear Second-Order ODE

The procedure for solving linear second-order ode has two steps

(1)

Find the general solution of the homogeneous problem:

According to the theory for linear differential equations, the general solution of the homogeneous problem is

where C1 and C2 are constants and y1 and y2 are any two linearly independent solutions to the homogeneous equation.

(2)

Find a particular solution of the nonhomogeneous problem:

The particular solution is any solution of the nonhomogeneous problem and is denoted yp(t).

The general solution of the full nonhomogeneous problem is

The key point to note is that all possible solutions to a linear second-order ode can be obtained from two linearly independent solutions to the homogeneous problem and any particular solution.

Here is an example. Consider the ode

The homogeneous equation is

It can be shown that y_1=exp(-t) and y_2=exp(-2t) are solutions to the homogeneous equation. Plug these expressions into the ode and verify!

A particular solution of the nonhomogeneous equation is exp(t). Hence, the general solution of the ode is

where C1 and C2 are constants.

Linear Dependence

Two functions linearly independent if they are not multiples of each other. For example, exp(-t) and exp(-2t) are linear independent. On the other hand, t+3 and 7t+21 are linearly dependent.

Techniques for Solving Homogeneous Linear Second-Order ODE

Certain classes of homogeneous linear second-order ode can be solved analytically. We will consider two classes:

(1)

Constant-coefficient ode:

p(t) and q(t) are constants.

(2)

Euler-Cauchy ode:

p and q are constants.

Techniques for Determining a Particular Solution

There are two principal techniques for determining a particular solution:

(1)

The Method of Undetermined Coefficients

(2)

Variation of Parameters

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