Math Problem Statement
what is the integration facto of dy / dx + 1 / x y= x square
Solution
To solve the differential equation , we can identify that this is a first-order linear differential equation of the form:
Here:
Step 1: Finding the integrating factor
The integrating factor is given by:
The integral of is , so:
Since is positive (due to the form of the problem), we can write:
Step 2: Multiply the entire equation by the integrating factor
Now, multiply the original differential equation by :
Simplifying:
Step 3: Solve the equation
The left-hand side can now be written as a derivative:
Now, integrate both sides with respect to :
The integral of is:
Step 4: Solve for
Now, divide by :
Thus, the solution to the differential equation is:
Would you like further clarification or more details on the process? Here are five related questions you might explore:
- How do we handle the integration factor for more complex functions?
- What happens if is a more complicated function, such as a trigonometric one?
- How does the integrating factor method compare to other methods for solving first-order linear differential equations?
- Can this method be extended to non-linear differential equations?
- How would the solution change if was instead of ?
Tip: Always verify the final solution by plugging it back into the original differential equation.
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Math Problem Analysis
Mathematical Concepts
First-Order Linear Differential Equations
Integrating Factors
Calculus
Formulas
First-order linear differential equation: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^∫P(x) dx
Theorems
Integrating Factor Method
Suitable Grade Level
Undergraduate Mathematics or Advanced High School Calculus
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