Linear First-Order Differential Equations

A linear first-order ode has the form:

where g(t) and h(t) are given functions. In addition to the solution procedure outlined below, it may also be possible to solve the problem using the Laplace Transform technique.

Solution Procedure

Rewrite the problem in the form:

We multiply the above equation by an integrating factor:

The derivative of u(t), which we will use below, is

(Recall the derivative of an exponential is the exponential multiplied by the
derivative of its argument. The argument is an integral, which we differentiate
by the Fundamental Theorem of Calculus.) The new equation is

Why did we do this? Look at the left-hand side of the equation. Let z(t)=u(t)y(t).
Using the product rule and the result above for u'(t), we have

Hence, equation (*) becomes

This is now in the form of a directly integrable equation since both u(t) and h(t)
are known. Applying the indefinite form of the Fundamental Theorem of Calculus,
we obtain