Solve the first-order linear differential equation dy/dx + y/x = 1.

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Question: Solve the first-order linear differential equation dy/dx + y/x = 1.

Options:

Correct Answer: y = x + C/x

Solution:

Using the integrating factor e^(∫(1/x)dx) = x, we solve to get y = x + C/x.

Solve the first-order linear differential equation dy/dx + y/x = 1.

Solve the first-order linear differential equation dy/dx + y/x = 1.

Step 1: Identify the differential equation: dy/dx + y/x = 1.
Step 2: Recognize that this is a first-order linear differential equation.
Step 3: Find the integrating factor. The integrating factor is e^(∫(1/x)dx).
Step 4: Calculate the integral: ∫(1/x)dx = ln|x|.
Step 5: Therefore, the integrating factor is e^(ln|x|) = x.
Step 6: Multiply the entire differential equation by the integrating factor (x): x(dy/dx) + y = x.
Step 7: Notice that the left side is the derivative of (y * x): d(y * x)/dx = x.
Step 8: Integrate both sides: ∫d(y * x) = ∫x dx.
Step 9: This gives us y * x = (1/2)x^2 + C, where C is the constant of integration.
Step 10: Solve for y: y = (1/2)x + C/x.

First-Order Linear Differential Equations – These equations can be expressed in the form dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x.
Integrating Factor – A function used to simplify the process of solving linear differential equations, calculated as e^(∫P(x)dx).
General Solution – The solution to a differential equation that includes an arbitrary constant, representing a family of solutions.