Problem 30 - Essential Calculus

Question

Solve the differential equation. Use a graphing utility to graph three solutions, one of which passes through the given point. $$ \frac{d r}{d t}=\frac{\sec ^{2} t}{\tan t+1}, \quad(\pi, 4) $$

Step-by-Step Solution

Answer

The solution to the differential equation is \( r = \ln|\tan\,t+1|+4 \).

Step 1 Separate variables

Separate variables by taking \( \sec^{2}\,t \) on the left side and \( \frac{dt}{\tan\,t+1} \) on the right side. So, we have \( dr= \frac{\sec^{2}\,t}{\tan\,t+1} \,dt \).

Step 2 Integrate both sides

Now, integrate both sides. The integration on the left side is direct, yielding \( r \). The integration on the right side is a simple substitution, with \( u = \tan\,t+1 \), \( du = \sec^{2}\,t \, dt \), yielding \( \int{du/u} \). We end up with \( r = \ln|u|+C \), which simplifies to \( r = \ln|\tan\,t+1|+C \).

Step 3 Determine the integration constant

Find the constant of integration using the given point \((\pi, 4)\). Substituting these values into the equation, we get \( 4 = \ln|\tan\,\pi+1|+C \). Since \( \tan\,\pi = 0 \), the constant \( C = 4 \).

Step 4 Write the final solution

Substitute \( C \) into the equation from Step 2 to get the final solution: \( r = \ln|\tan\,t+1|+4 \).