Find the solution of the differential equation dr/dt = {sec^2 t}/{tan t + 1} which passes through...


Find the solution of the differential equation {eq}\frac{\displaystyle dr}{\displaystyle dt} = \frac{\displaystyle sec^2 t}{\displaystyle \tan t + 1} {/eq} which passes through the point {eq}(\pi, 5) {/eq}.

Answer and Explanation:

To solve the problem we will separate the variables:

{eq}\frac{\mathrm{d} r}{\mathrm{d} t}=\frac{\sec^{2}t}{\tan t+1} {/eq}

Now using the substitution method we will solve the problem:

{eq}\tan t+1=x\\ \sec^{2}tdt=dx {/eq}

Now the integral becomes:

{eq}\int dr=\int \frac{dx}{x}\\ r=\ln x+c {/eq}

Now by back substitution:

{eq}r=\ln (\tan t+1)+c\\ t=\pi, r=5\\ 5=c\\ r=\ln (\tan t+1)+5 {/eq}

Learn more about this topic:

Separable Differential Equation: Definition & Examples

from GRE Math: Study Guide & Test Prep

Chapter 16 / Lesson 1

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