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Separable Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \frac{y^2 - 9y + 20}{2x + 4}\\ y(0) &= 2 \end{align}$$This is a separable equation. First we find the general solution following the paradigm.

  1. Separate the variables $$ \frac{dy}{y^2 - 9y + 20} = \frac{dx}{2x + 4} $$
  2. Integrate both sides $$ \log|y - 5| - \log|y - 4| = (1/2)\log|2x + 4| + C $$
  3. Solve for y (if possible)

    We simplify the expression to obtain the general solution $$ \left(\frac{y - 5}{y - 4}\right)^2 = k (2x + 4) $$

  4. Check for singular solutions

    We divided by $y^2 - 9y + 20$ which is 0 when $y = 5$ or $y = 4.$ We can check these are both solutions. Now $y = 5$ is already in the general solution (when the constant is 0), but $y = 4$ is a singular solution.

Now we plug in the initial values $x = 0$ and $y = 2$ and solve for the arbitrary constant, which we compute to be $9/16.$ So the solution to the initial value problem is $$\left(\frac{y - 5}{y - 4}\right)^2 = (9/16) (2x + 4).$$

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