Consider the linear differential equation of the form
where f and g are functions of t and is a constant. We assume that g(t) is given and we wish to find f(t).
To solve such equations, we make the left side have the form of the derivative of a product by multiplying through by :
We recognize the left as the derivative of the product of and f:
We can integrate both sides of this (with s substituted for t) from 0 to t to recover f(t):
Of course, the integral of the derivative on the left side of the equation may be evaluated using the (second) fundamental theorem of calculus:
Now we can solve algebraically for f(t):