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)

\begin{align} y = De^{at} - \frac{b}{a} \end{align}

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)

\begin{align} \quad \frac{dy}{dt} = -100(y - 0.5) \\ \quad \frac{\frac{dy}{dt}}{y - 0.5} = -100 \\ \quad \frac{d}{dt} \ln \mid y - 0.5 \mid = -100 \\ \quad \int \frac{d}{dt} \ln \mid y - 0.5 \mid = \int -100 \: dt \\ \quad \ln \mid y - 0.5 \mid = -100t + C \\ \quad \mid y - 0.5 \mid = e^{-100t + C} \\ \quad y - 0.5 = \pm e^{-100t}e^C \\ \quad y = De^{-100t} + 0.5 \end{align}

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)

\begin{align} \quad \frac{dy}{dt} = 55 \left ( y - \frac{17}{55} \right ) \\ \quad \frac{\frac{dy}{dt}}{y - \frac{17}{55}} = 55 \\ \quad \frac{d}{dt} \ln \biggr \rvert y - \frac{17}{55} \biggr \rvert = 55 \\ \quad \int \frac{d}{dt} \ln \biggr \rvert y - \frac{17}{55} \biggr \rvert \: dt = \int 55 \: dt \\ \quad \ln \biggr \rvert y - \frac{17}{55} \biggr \rvert = 55t + C \\ \quad \biggr \rvert y - \frac{17}{55} \biggr \rvert = e^{55t + C} \\ \quad y - \frac{17}{55} = \pm e^{55t}e^{C} \\ \quad y = De^{55t} + \frac{17}{55} \end{align}

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.