Math Problem Statement
Find the general solution of the differential equation. Then, use the initial condition to find the corresponding particular solution. xy'+3y=4x, y(1)=6
Solution
We are given the first-order linear differential equation:
and the initial condition .
Step 1: Rewrite the equation in standard form
First, we divide the entire equation by to rewrite it in standard form:
This is a linear first-order differential equation of the form , where:
Step 2: Find the integrating factor
The integrating factor is given by:
Since , we can simplify .
Step 3: Multiply the equation by the integrating factor
We now multiply the entire equation by the integrating factor :
The left-hand side is the derivative of :
Step 4: Integrate both sides
Integrating both sides with respect to :
where is the constant of integration. Thus, the general solution is:
Step 5: Apply the initial condition
We are given . Substituting and into the general solution:
Solving for :
Step 6: Write the particular solution
Substitute into the general solution to get the particular solution:
Thus, the particular solution is:
Would you like to explore any of the steps in more detail, or do you have any questions?
Here are 5 related questions for further practice:
- Solve with .
- Solve with .
- Find the general solution of .
- Solve the differential equation .
- Solve with the initial condition .
Tip: For linear first-order differential equations, always look for an integrating factor to simplify the equation.
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Math Problem Analysis
Mathematical Concepts
Differential Equations
Linear First-Order Differential Equations
Integrating Factor
Formulas
Standard form of a linear first-order differential equation: y' + P(x)y = Q(x)
Integrating factor: μ(x) = e^∫P(x) dx
General solution: y = (x^4 + C)/x^3
Theorems
Method of Integrating Factors
Suitable Grade Level
Undergraduate - Differential Equations Course
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