# The Method of Direct Integration Examples 1

Recall from The Method of Direct Integration page that if we have a differential equation in the form $\frac{dy}{dt} = ay + b$ where $a$ and $b$ are constants, then the general solution can be obtained by rewriting this differential equation as to get the lefthand side to be the derivative of a natural logarithm while in general, the solution is given as:

(1)

Let’s look at some examples of solving differential equations by direct integration.

## Example 1

**Find all solutions to the differential equation $\frac{dy}{dt} = -100y + 50$.**

The differential equation above can be rewritten for $y \neq 0.5$ and solved as follows:

(2)

Note that if $D = 0$ then $y = 0.5$ which is a solution to the differential equation $\frac{dy}{dt} = -100y + 50$ and so for all $D$ as constants, $y = De^{-100t} + 0.5$ gives all solutions.

## Example 2

**Find all solutions to the differential equation $\frac{dy}{dt} = 55y – 17$.**

The differential equation above can be rewritten for $y \neq \frac{17}{55}$ and solved as follows:

(3)

Note that if $D = 0$ then $y = \frac{17}{55}$ which is a solution to the differential equation $\frac{dy}{dt} = 55y – 17$ and so for all $D$ as constants, $y = De^{55t} + \frac{17}{55}$ gives all solutions.