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Chapter 4: Problem 29

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

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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 \( C = 2 - \frac{1}{2}\ln |1| = 2 \).
04

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 \( s( \theta) = \frac{1}{2}\ln |\sec(2 \theta)| + 2 \).

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Most popular questions from this chapter

In Exercises 87-89, consider a particle moving along the \(x\) -axis where \(x(t)\) is the position of the particle at time \(t, x^{\prime}(t)\) is its velocity, and \(\int_{a}^{b}\left|x^{\prime}(t)\right| d t\) is the distance the particle travels in the interval of time. The position function is given by \(x(t)=t^{3}-6 t^{2}+9 t-2\) \(0 \leq t \leq 5 .\) Find the total distance the particle travels in 5 units of time.

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