Math 240 Home, Textbook Contents, Online Homework Home

Warning: MathJax requires JavaScript to process the mathematics on this page.
If your browser supports JavaScript, be sure it is enabled.

First Order Linear Equations

Additional Examples

Solve the following initial value problem $$ \begin{align} \frac{dy}{dx} + 9 y &= 3\exp(-7 x) \\ y(0) &= 7 \end{align} $$ This is a linear equation. First we find the general solution following the paradigm.

  1. Find the integrating factor $$ \mu(x) = \exp(\int 9 dx) = \exp(9x) $$
  2. Multiply through by the integrating factor $$ \exp(9x) \frac{dy}{dx} + 9\exp(9x)y = 3\exp(-7 x)\exp(9x) = 3\exp(2x) $$
  3. Recognize the left-hand-side as $\displaystyle \frac{d}{dx}(\mu(x)y).$ $$ \frac{d}{dx}(\exp(9x)y) =3\exp(2x) $$
  4. Integrate both sides. $$ \exp(9x)y = (3/2)exp(2x) + C $$
  5. Divide through by $\mu(x)$ to solve for $ y.$ $$y = (3/2)exp(-7x) + C\exp(-9x) $$
Now we plug in the initial values $ x = 0 $ and $ y = 7$ and solve for $ C = 11/2$, to obtain the solution to the initial value problem $$ y = (3/2)\exp(-7x) + (11/2)\exp(-9x) $$ You may reload this page to generate additional examples.


If you have any problems with this page, please contact bennett@math.ksu.edu.
©2010, 2014 Andrew G. Bennett