Mathematics Department

Textbook Contents

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Separable Equations

Additional Examples

Solve the following initial value problem, $$\begin{align} \frac{dy}{dx} &= 5\cos(3x)\exp(-3y)\\ y(\pi/18) &= 1 \end{align}$$This is a separable equation. First we find the general solution following the paradigm.

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

    We simplify the expression to obtain the general solution $$ y = (1/3)\log(5\sin(3x) + C) $$

  4. Check for singular solutions

    Since we never divided by anything that could be 0, there are no singular solutions.

Now we plug in the initial values $x = \pi/18$ and $y = 1$ and solve for the arbitrary constant, which we compute to be $\exp(3) - 5/2.$ So the solution to the initial value problem is $$y = (1/3)\log(5\sin(3x) + \exp(3) - 5/2).$$

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