As so btw "Differential equations 1 2 3" likely refers to different levels of a differential equations course. 1st-order differential equations are solved using methods like separation of variables or integrating factors. 2nd-order equations often involve characteristic equations and solving for roots to find solutions, as seen in the form \(ay^{\prime \prime }+by^{\prime }+cy=0\). While there isn't a standard "3" for a basic level, it could refer to systems of equations or a third-order equation, though these are typically introduced in more advanced courses.
Why are integrating factors useful for solving first-order linear differential equations?
because they transform the equation into a form where the left side becomes the derivative of a product, which can then be easily integrated. By multiplying the entire equation by the integrating factor, a non-exact equation is rewritten as an exact one, simplifying the solution process and allowing for a straightforward integration to find the solution (Y,X).To make the left side match this form, you need \(\mu ^{\prime }(x)=\mu (x)p(x)\). This is a separable differential equation that can be solved to find the integrating factor, which is \(\mu (x)=e^{\int p(x)dx}\)
