Linear First-Order Differential Equations

The differential equation of the form is a linear first-order differential eqution. The solution to this equation is given by the formula

. That this is indeed a solution of the differential equation is easily checked using Maple.

> y :=x-> (int(f(x)*exp(int(a(x),x)),x)+C)*exp(-int(a(x),x));

diff(y(x),x)+a(x)*y(x)-f(x);

> simplify(%);

We will use this formula to solve first-order linear equations. To faciliate are calulations we will introduce an operator FO (FO stands for first-order), and in this operator we will be substituting and .

> FO:=(int(f(x)*exp(int(a(x),x)),x)+C)*exp(-int(a(x),x));

Example 1: Solve .

Example 2: Solve where .