Math Problem Statement
Solution
The initial value problem given is:
Step 1: Solve the differential equation
We start by separating the variables to integrate both sides:
Integrating both sides with respect to :
The integral of is , so:
Step 2: Apply the initial condition
We are given that . Substituting into the solution:
Final Solution:
Thus, the solution to the initial value problem is:
Let me know if you would like further details or have any questions.
Here are 5 questions to further explore this topic:
- What is the general method for solving separable differential equations?
- How does the initial condition influence the solution of a differential equation?
- What other types of functions could replace in the differential equation?
- Can this method be applied to non-homogeneous differential equations?
- How does the periodicity of 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)
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