Mathematics Department

Math 340 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.

Separable Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{y + 4}{x^2 + 3x + 2}\\ y(1) &= 4 \end{align}$$This is a separable equation. First we find the general solution following the paradigm.

  1. Separate the variables $$ \frac{dy}{y + 4} = \frac{dx}{x^2 + 3x + 2} $$
  2. Integrate both sides $$ \log|y + 4| = \log|x + 1| - \log|x + 2| + C $$
  3. Solve for y (if possible)

    We simplify the expression to obtain the general solution $$ y + 4 = k \frac{x + 1}{x + 2} $$

  4. Check for singular solutions

    We divided by $y + 4$ which is 0 when $y = -4$, which we can check is a solution. Note this solution is already in the general solution (when the constant is 0).

Now we plug in the initial values $x = 1$ and $y = 4$ and solve for the arbitrary constant, which we compute to be $12.$ So the solution to the initial value problem is $$y + 4 = 12 \frac{x + 1}{x + 2}.$$

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