First-Order Linear Equations
Solve the differential equations in Exercises 1–14.


tan θ dr/dθ + r = sin²θ, 0 < θ < π/2

Verified step by step guidance

1. Rewrite the given differential equation in the standard linear form. The original equation is \(\tan \theta \frac{dr}{d\theta} + r = \sin^{2} \theta\). Divide both sides by \(\tan \theta\) to isolate \(\frac{dr}{d\theta}\), resulting in \(\frac{dr}{d\theta} + \frac{r}{\tan \theta} = \frac{\sin^{2} \theta}{\tan \theta}\).
2. Identify the integrating factor (IF) for the linear differential equation \(\frac{dr}{d\theta} + P(\theta) r = Q(\theta)\), where \(P(\theta) = \frac{1}{\tan \theta}\). The integrating factor is given by \(\mu(\theta) = e^{\int P(\theta) d\theta}\).
3. Calculate the integral \(\int \frac{1}{\tan \theta} d\theta\). Recall that \(\frac{1}{\tan \theta} = \cot \theta\), so the integral becomes \(\int \cot \theta d\theta\). Use the known integral formula for \(\cot \theta\).
4. Multiply the entire differential equation by the integrating factor \(\mu(\theta)\) to write the left side as the derivative of the product \(\mu(\theta) r\). This transforms the equation into \(\frac{d}{d\theta} [\mu(\theta) r] = \mu(\theta) Q(\theta)\).
5. Integrate both sides with respect to \(\theta\) to find \(\mu(\theta) r = \int \mu(\theta) Q(\theta) d\theta + C\). Finally, solve for \(r(\theta)\) by dividing both sides by \(\mu(\theta)\).

Key Concepts

First-Order Linear Differential Equations

These are differential equations of the form dy/dx + P(x)y = Q(x), where the highest derivative is first order and the equation is linear in the unknown function and its derivative. Solving them typically involves finding an integrating factor to simplify the equation into an exact derivative.

Integrating Factor Method

An integrating factor is a function, usually denoted μ(x), used to multiply a linear differential equation to make the left side an exact derivative. It is found by μ(x) = e^(∫P(x)dx), enabling straightforward integration and solution of the equation.

Trigonometric Functions and Their Properties

Understanding the behavior and identities of trigonometric functions like tan(θ) and sin²(θ) is essential. These functions often appear in differential equations involving angular variables, and their properties help simplify and integrate terms effectively.