Here are the steps to solve a first-order linear differential equation:

Write the differential equation in the form:

dy/dx + P(x)y = Q(x)

where P(x) and Q(x) are functions of x, and y is the unknown function we want to find.

Find the integrating factor u(x), which is defined as:

u(x) = e^(integral of P(x) dx)

where the integral is taken with respect to x.

Multiply both sides of the differential equation by the integrating factor u(x):

u(x) dy/dx + u(x)P(x)y = u(x)Q(x)

Use the product rule of differentiation to rewrite the left-hand side of the equation:

(u(x)y)' = u(x)Q(x)

where (u(x)y)' denotes the derivative of u(x)y with respect to x.

Integrate both sides of the equation with respect to x:

u(x)y = integral of u(x)Q(x) dx + C

where C is the constant of integration.

Solve for y:

y = (1/u(x)) * (integral of u(x)Q(x) dx + C)

Simplify the expression for y by evaluating the integral and substituting the value of u(x) from step 2.

That's it! These steps should allow you to solve any first-order linear differential equation.
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