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

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

Step 1: Identify the differential equation: dy/dx + y/x = x.
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|. Since x is positive in this context, we use x.
Step 6: Multiply the entire differential equation by the integrating factor x: x(dy/dx) + y = x^2.
Step 7: Notice that the left side is the derivative of (y * x): d/dx(y * x) = x^2.
Step 8: Integrate both sides: ∫d/dx(y * x) dx = ∫x^2 dx.
Step 9: This gives us y * x = (1/3)x^3 + C, where C is the constant of integration.
Step 10: Solve for y: y = (1/3)x^2 + C/x.

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