Math Problem Statement
Solution
The image contains differential equations that need to be solved for with the initial condition . Let's break down and solve each one:
Part a)
This is a first-order linear differential equation. Let's solve it step by step:
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Rewrite the equation:
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This can be solved using the separation of variables technique. Separate and as follows:
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Integrating both sides:
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Solve for :
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Apply the initial condition : Therefore, , and the final solution is:
Part b)
This is of the same form as part (a). Following the same steps:
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Rewrite:
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Separate variables and integrate:
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Apply the initial condition :
Part c)
Again, this is a first-order linear differential equation. Follow the same method as in parts (a) and (b):
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Rewrite:
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Separate variables and integrate:
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Apply the initial condition :
Summary of Solutions:
- (a)
- (b)
- (c)
These are standard solutions for first-order linear differential equations with constant coefficients.
Would you like further details or explanations on any of these solutions?
Here are 5 follow-up questions to expand your understanding:
- How do you solve non-homogeneous first-order differential equations?
- What is the physical meaning of these differential equations in electrical circuits?
- How would you approach solving a second-order differential equation?
- What is the significance of the exponential term in the solutions?
- How does the method of integrating factors apply to solving linear differential equations?
Tip: When solving first-order differential equations, always try separation of variables first—it simplifies many common problems!
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Linear First-Order Differential Equations
Separation of Variables
Formulas
a·dy/dt + p·y = 0
m·dy/dt + b·y = 0
R·C·dy/dt + y = 0
Theorems
Solution of Linear First-Order Differential Equations
Exponential Decay
Suitable Grade Level
Undergraduate (First-Year Calculus or Differential Equations Course)
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