Math Problem Statement
Please explain how they got yhe answer
Solution
The problem asks for the solution of the differential equation:
Step-by-step Explanation:
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Rewrite the differential equation: Start by simplifying the given equation: Rearrange it: Now, divide both sides by to isolate :
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Separate variables: The next step is to use separation of variables. Multiply both sides by and to separate and :
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Integrate both sides: Now, integrate both sides of the equation. The left-hand side is straightforward: For the right-hand side: Using the substitution , with , the integral becomes: So, after integration, we get:
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Solve for : Multiply both sides by 2 to simplify:
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Use the initial condition : To find the constant , use the initial condition . Substitute and into the equation: Solving for :
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Final solution: Substitute the value of back into the equation:
This is the final solution to the differential equation.
Would you like more details on any step or do you have any questions?
Related Questions:
- What substitution was used to integrate the right-hand side of the equation?
- Why is separation of variables a useful technique for solving this type of differential equation?
- How would the solution change if the initial condition was different, e.g., ?
- Can this equation be solved using any other methods, such as an integrating factor?
- What is the importance of the initial condition in determining the constant ?
Tip:
When integrating a function that includes an exponential and a polynomial, substitution often simplifies the process effectively.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Separation of Variables
Initial Value Problem
Formulas
y * dy = -x * e^(-x^2) * dx
∫ y dy = ∫ -x * e^(-x^2) dx
Substitution method: u = x^2
Theorems
Separation of Variables Technique
Initial Condition Application
Suitable Grade Level
Undergraduate level (Calculus II or Differential Equations course)
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