Math Problem Statement
Do question 23 and show the work please
Solution
Let's solve question 23:
Given the differential equation:
with the initial condition .
Step 1: Separate Variables
Separate and to integrate each side independently.
Step 2: Integrate Both Sides
Now integrate both sides with respect to their respective variables.
The left side integrates to:
The right side integrates to:
So we have:
Step 3: Multiply Both Sides by 2 to Simplify
Let , where is a new constant, giving us:
Step 4: Apply the Initial Condition
Using the initial condition :
So the particular solution becomes:
Step 5: Solve for
To get explicitly, take the square root of both sides:
Given , we select the positive branch:
Final Solution
The solution to the differential equation is:
Would you like further details on any of the steps, or have any questions?
Here are some related questions that might help deepen your understanding:
- How would the solution change if the initial condition were different?
- What happens if we select the negative branch of the square root?
- Can this technique be applied to other differential equations of similar form?
- What is the significance of separating variables in differential equations?
- How does the initial condition affect the constant of integration?
Tip: Always check if you can separate variables directly when solving differential equations, as it simplifies the process considerably!
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Separation of Variables
Initial Value Problems
Formulas
\( \frac{dy}{dx} = \frac{x^3}{y} \)
Separation of variables technique
Integration of \( \int y \, dy = \int x^3 \, dx \)
Theorems
Initial Value Theorem
Suitable Grade Level
Undergraduate
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