Problem 29 - Essential Calculus

Question

Solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. $$ \frac{d s}{d \theta}=\tan 2 \theta, \quad(0,2) $$

Step-by-Step Solution

Answer

The solution of the differential equation \( \frac{d s}{d \theta}=\tan 2 \theta \) through the point (0,2) is \( s(\theta) = \frac{1}{2}\ln |\sec(2 \theta)| + 2 \).

Step 1: Separate Variables

In order to solve this differential equation, we would start by separating the variables by moving the \( d\theta \) to the right-hand side resulting in \( ds = \tan(2\theta) d\theta \). This makes it possible to integrate both sides.

Step 2: Integrate both sides

Now we can integrate both sides. The left side is very straightforward, while for the right side we recognize the integral of the tangent function. It results in \( s = \frac{1}{2}\ln |\sec(2\theta)| + C \), where C is the constant of integration.

Step 3: Apply the given point

We can find the value of C using the given point (0,2), so substituting \( \theta = 0 \) and \( s = 2 \) : \( 2 = \frac{1}{2}\ln |\sec(2*0)| + C \) This gives \( C = 2 - \frac{1}{2}\ln |1| = 2 \).

Step 4: Write the final form of the solution

After finding the value of the integration constant C, we can substitute it back into the solution of the differential equation. Therefore, the solution of the differential equation that passes through the point (0,2) is \( s( \theta) = \frac{1}{2}\ln |\sec(2 \theta)| + 2 \).