How do I integrate an inseparable equation?

Question

 I'm confused on this topic in my Diff Eq class.

Answer 1

First, let's discuss separable equations.   Separable differential equations are in the form

 

Mdy - Ndx = 0

 

Mdy = Ndx

 

 

In which  M is a function of y, and N is a function of x.  When we solve for y, we can easily integrate both sides of the equation because integrals and derivatives cancel each other out, leaving only the functions on both sides of the equation by themselves.

 

When we manipulate this equation, we obtain  dy/dx = N/M

 

 

This newly formed equation is a first-order differential equation, which does not have a second derivative term nor higher .  Therefore, only first-order differential equations are separable. 

 

In addition to this, if the equation is only in terms of y, then the equation is not separable although it is first-order.  This is because separable equations have x and y variables, as shown in earlier.  So you would need to use the integrating factor for first-order linear equations, and Bernoulli method for first-order non-linear equations.

 

Knowing all of this, we now know that inseparable equations are second-order and higher.

 

For second-order equations, we use a characteristic equation to find the roots, and plug them into a general form of the solution.   You may recall solving homogenous equations that has at most a second derivative term.

 

 

For equations of higher-order, there are advanced methods, such as Euler's Method and Runge Kutta.  You will learn that later in the course.

 

 

 

 

I hope this helps!!