Math Problem Statement
dy/dx=-1-3y
Solution
The given differential equation is:
This is a first-order linear differential equation. To solve it, we will use the method of separation of variables or an integrating factor. Let's proceed with the integrating factor method.
Step 1: Rewrite the equation
This is now in the standard linear form:
where and .
Step 2: Find the integrating factor
The integrating factor is given by:
Step 3: Multiply the equation by the integrating factor
Multiplying both sides of the equation by , we get:
The left-hand side is now the derivative of :
Step 4: Integrate both sides
Integrating both sides with respect to :
The integral on the right side is:
Thus, we have:
Step 5: Solve for
To solve for , divide both sides by :
Thus, the general solution to the differential equation is:
where is the constant of integration.
Would you like more details on any step or have any questions?
Here are some related questions for further exploration:
- What is the general form of a linear differential equation?
- How does the integrating factor method work for solving differential equations?
- Can this differential equation be solved using separation of variables?
- What are the conditions for a differential equation to be classified as first-order linear?
- How can initial conditions be applied to determine the constant ?
Tip: When dealing with first-order linear differential equations, the integrating factor method is a reliable approach to finding the solution!
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Math Problem Analysis
Mathematical Concepts
Differential Equations
First-order Linear Equations
Integrating Factor
Formulas
First-order linear form: dy/dx + P(x)y = Q(x)
Integrating factor: μ(x) = e^∫P(x)dx
Theorems
Integrating Factor Method
Suitable Grade Level
Undergraduate level (Calculus I or II)
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