Math Problem Statement

Solve the following initial value problem: dr/dθ = -πsin(πθ), r(2) = 3.

Solution

The initial value problem given is:

drdθ=πsin(πθ),r(2)=3\frac{dr}{d\theta} = -\pi \sin(\pi \theta), \quad r(2) = 3

Step 1: Solve the differential equation

We start by separating the variables to integrate both sides:

drdθ=πsin(πθ)\frac{dr}{d\theta} = -\pi \sin(\pi \theta)

Integrating both sides with respect to θ\theta:

r(θ)=πsin(πθ)dθr(\theta) = \int -\pi \sin(\pi \theta) d\theta

The integral of sin(πθ)\sin(\pi \theta) is 1πcos(πθ)-\frac{1}{\pi} \cos(\pi \theta), so:

r(θ)=π(1πcos(πθ))+Cr(\theta) = -\pi \left( -\frac{1}{\pi} \cos(\pi \theta) \right) + C r(θ)=cos(πθ)+Cr(\theta) = \cos(\pi \theta) + C

Step 2: Apply the initial condition

We are given that r(2)=3r(2) = 3. Substituting θ=2\theta = 2 into the solution:

r(2)=cos(π×2)+C=cos(2π)+C=1+Cr(2) = \cos(\pi \times 2) + C = \cos(2\pi) + C = 1 + C 3=1+C3 = 1 + C C=2C = 2

Final Solution:

Thus, the solution to the initial value problem is:

r(θ)=cos(πθ)+2r(\theta) = \cos(\pi \theta) + 2

Let me know if you would like further details or have any questions.


Here are 5 questions to further explore this topic:

  1. What is the general method for solving separable differential equations?
  2. How does the initial condition influence the solution of a differential equation?
  3. What other types of functions could replace sin(πθ)\sin(\pi \theta) in the differential equation?
  4. Can this method be applied to non-homogeneous differential equations?
  5. How does the periodicity of cos(πθ)\cos(\pi \theta) affect the behavior of the solution?

Tip: Always ensure to apply the initial conditions carefully after integrating to solve for any constants of integration.

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Math Problem Analysis

Mathematical Concepts

Differential Equations
Initial Value Problems
Trigonometric Functions

Formulas

dr/dθ = -πsin(πθ)
Integral of sin(πθ) = -(1/π)cos(πθ)

Theorems

Separation of Variables
Initial Conditions in Differential Equations

Suitable Grade Level

College Level (Calculus II or Differential Equations)