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Separable Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= 7\cos(2x)(y + 5)\\ y(\pi/12) &= 4 \end{align}$$This is a separable equation. First we find the general solution following the paradigm.

  1. Separate the variables $$ \frac{dy}{y + 5} = 7\cos(2x) dx $$
  2. Integrate both sides $$ \log|y + 5| = (7/2)\sin(2x) + C $$
  3. Solve for y (if possible)

    We simplify the expression to obtain the general solution $$ y + 5 = k \exp((7/2)\sin(2x)) $$

  4. Check for singular solutions

    We divided by $y + 5$ which is 0 when $y = -5$, 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 $x = \pi/12$ and $y = 4$ and solve for the arbitrary constant, which we compute to be $9\exp(-7/4).$ So the solution to the initial value problem is $$y + 5 = 9\exp((7/2)(\sin(2x) - 1/2)).$$

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