Math Problem Statement
y'=-2y+5
Solution
You have the differential equation:
This is a first-order linear ordinary differential equation. To solve it, we can use the method of separation of variables or integrating factor.
Let's proceed with the integrating factor method. The equation can be rewritten in the form:
This is a linear equation of the form:
where and . The solution method involves finding the integrating factor , given by:
Multiply the entire differential equation by the integrating factor :
Now, the left side of the equation is the derivative of :
Next, integrate both sides with respect to :
The integral of the left-hand side is , and the integral of the right-hand side is:
So, we have:
Where is the constant of integration. Now, divide through by :
This is the general solution to the differential equation.
Would you like more details on any step, or have any further questions on this?
Here are some related questions to explore:
- How does the integrating factor method work for first-order linear differential equations?
- What does the constant represent in the general solution?
- How can we solve this type of equation using separation of variables?
- What happens if the right-hand side of the equation is a more complex function (not just a constant)?
- Can you solve a similar equation where the coefficient of is not constant (e.g., )?
Tip: When solving differential equations, always check if the equation is linear or separable to choose the best solving method.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
First-Order Linear Differential Equations
Integrating Factor Method
Formulas
dy/dx + P(x)y = Q(x)
Integrating Factor: μ(x) = e^{∫P(x)dx}
General Solution: y = (1/μ(x)) ∫(μ(x) Q(x) dx) + C/μ(x)
Theorems
Integrating Factor Theorem
Suitable Grade Level
Grades 11-12, College Calculus
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