Solving equations of the form dy/dx + Py = Q using an integrating factor

Derivation and Understanding of Differential Equations

First-order linear differential equations often occur in our everyday lives, with examples ranging from exponential growth and decay (populations, radioactive matter) to cooling and heating laws in physics.
The general form for such equations is dy/dx + P(x)y = Q(x), where P and Q are functions of x.
To solve this type of equation, we introduce an integrating factor. This is an exponential function whose purpose is to make the left-hand side of the equation a perfect derivative. Its general form is e^∫P(x)dx.

Procedure for Solving First-Order Linear Differential Equations

Start by identifying the coefficient of y in the given equation, labeled P(x).
Evaluate the integral of P(x) with respect to x, i.e., ∫P(x)dx. This calculation does not require the constant of integration.
The exponential of this integral, e^∫P(x)dx, gives us the integrating factor.
Multiply every term in the equation by this integrating factor.
The result will be a product rule form of a derivative. The equation can then be written as d/dx[{Integrating Factor}y] = {Integrating Factor}Q(x).
Integrate both sides of this modified equation with respect to x.
The left side of the equation would leave you with {Integrating Factor}*y, as the integral of a derivative is the function itself.
Performing the integration on the right side might require techniques such as simple antiderivatives, substitution, partial fractions, etc., depending on the nature of the function.
Finally, divide both sides of the equation by the integrating factor, leaving you with the solution for y in terms of x (i.e., y(x)).

Practice and Application

By mastering this technique, a wide range of practical scenarios can be modelled and understood.
Practice using this systematic approach and various integration techniques to efficiently solve first-order linear differential equations.
Look for opportunities to apply this knowledge, for instance, to track populations of species, analyse electrical and mechanical systems, or study economic growth models.

Keep revising these steps and practising with different example equations. Ensure you understand the process - how and why it works. This will be crucial to successfully manoeuvre more complex calculus problems.