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Separable Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= \sin(3x)(y + 2)\\ y(\pi/12) &= 2 \end{align}$$This is a separable equation. First we find the general solution following the paradigm.

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

    We simplify the expression to obtain the general solution $$ y + 2 = k \exp(-(1/3)\cos(3x)) $$

  4. Check for singular solutions

    We divided by $y + 2$ which is 0 when $y = -2$, 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 $y = 2$ and solve for the arbitrary constant, which we compute to be $4\exp(\sqrt{2}/6).$ So the solution to the initial value problem is $$y + 2 = 4\exp(-(1/3)(\cos(3x) - \sqrt{2}/2)).$$

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