Final answer:
The student's question is regarding the solution of differential equations using integrating factors. An integrating factor, calculated as e²∫(∫(Pdx)), is multiplied by the differential equation, transforming it into a form where the left side is the derivative of the product of the integrating factor and y, leading to a simple integration to solve for y.
Step-by-step explanation:
The student is asking how to solve various differential equations using integrating factors. Integrating factors are a technique applied to first-order linear differential equations, which involves multiplication by an appropriately chosen function, the integrating factor. The method transforms a non-exact differential equation into an exact one, simplifying the process of integration.
For an equation of the form dy/dx + Py = Q, the integrating factor, often denoted as μ(x), is found by computing e²∫(Pdx). Multiplying the entire differential equation by this integrating factor allows us to rewrite the left side as the derivative of a product of the integrating factor and the unknown function y, which can then be easily integrated with respect to x. After integrating, the solution for y can be found, often involving the integration of Q multiplied by the integrating factor.