Math Problem Statement
Solve y' + y = 3x as a diff eq
Solution
The differential equation is:
This is a first-order linear differential equation. We can solve it using an integrating factor. Here's the step-by-step process:
Step 1: Write the equation in standard form
The standard form of a first-order linear differential equation is:
In this case, we can identify:
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 differential equation by :
Step 4: Recognize the left-hand side as a derivative
The left-hand side of the equation can now be written as the derivative of the product :
Step 5: Integrate both sides
Now, integrate both sides with respect to :
We can use integration by parts to solve the integral . Let:
- , so ,
- , so .
Now apply the integration by parts formula:
This gives:
Thus, we have:
Step 6: Solve for
Divide both sides by :
Final solution:
The general solution to the differential equation is:
where is an arbitrary constant determined by initial conditions if provided.
Do you have any questions or would you like further details?
Related questions:
- What are integrating factors, and how do they work in solving differential equations?
- Can you show how to apply integration by parts more clearly?
- What are some examples of initial conditions that could specify here?
- How can this method be extended to non-linear differential equations?
- Are there any alternative methods to solve this type of equation?
Tip:
When solving first-order linear differential equations, always check if the equation is in standard form to apply the integrating factor method easily.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Differential Equations
First-order Linear Differential Equations
Integration by Parts
Integrating Factors
Formulas
Standard form of first-order linear differential equations: y' + P(x)y = Q(x)
Integrating factor: μ(x) = e^{∫P(x) dx}
Integration by parts: ∫u dv = uv - ∫v du
Theorems
First-order Linear Differential Equation Theorem
Integration by Parts
Suitable Grade Level
Undergraduate Calculus (Grade 12 or College-level)
Related Recommendation
Solving First-Order Linear Differential Equation y'(x) - 3y(x) = x
Solving First-Order Linear Differential Equation y' + y = x
Solving First-Order Linear Differential Equation y' = x + y using Integrating Factors
Solve First-Order Linear Differential Equation y' + y = e^x
Solve First-Order Linear Differential Equation: dy/dx + 3y = 3x^3 - 4x^2 - 5x