First Order Linear Differential Equation

(1)

y' + p(t)y = f(t).

Both p(t) and f(t) are continuous functions of t, where t varies over some interval I in the real line. When f(t) is the constant function with value 0, the equation is a homogeneous linear differential equation of the type studied in Section 14.2.

First order linear differential equations arise in models of population growth with immigration. Suppose a population y(t) has a net birthrate of b(t) and net immigration rate of f(t). The net birthrate b(t) is the excess of births over deaths per unit of population in one unit of time. In a small period of time of length Δf, the difference of births and deaths is b(t) · y(t) · Δt, and the net immigration is f(t) · Δt. Then the population will be a solution of the differential equation y' = b(t)y + f(t), which is the same as equation (1) with p(t) = -b(t).

The size of a bank account that earns interest and also changes due to deposits and withdrawals can be described by a first order linear differential equation. If the account earns interest at the rate of r(t) at time t, and the net deposit per unit of time is f(t), then the account size y(t) will be a solution of the differential equation (1) with p(t) = -r(t).

The next theorem will be helpful in solving an equation of the type (1).

We already know from Section 14.2 how to solve the homogeneous linear equation (2). So if we can find one particular solution of the linear equation (1), we can use Theorem 1 to find the general solution. We postpone the proof of Theorem 1 to the end of this section.

A particular solution of a linear equation (1) can be found by the method called variation of constants. Start with a particular solution x(t) of the corresponding homogeneous equation (2). For any constant C, Cx(t) is also a solution of (2). Now replace the constant C by a variable u(t), and see what happens. Let y(t) = v(t)x(t). We shall compute the left side of equation (1), y' + p(t)y. If it turns out to be equal to f(t), then y(t) will be a particular solution of (1) as required. We carry out the computations using the Product Rule for derivatives.

y = vx.

y' + py = (vx)' + pvx

= v'x + vx' + pvx = v'x + v(x' + px).

Since x is a solution of the homogeneous equation (2),

x' + px = 0.

Therefore

y' + py = v'x.

Thus if we can find a function v(t) such that

v'(t)x(t) =f(t),

then

y(t) = v(t)x(t)

is a particular solution of the linear equation (1).

Putting all the ideas together, we have a method for solving a first order linear differential equation.