Virtual Lessons: Homogeneous DEs

Homogeneous differential equations are a type of differential equation in which all the terms involve only the dependent variable and its derivatives. These equations are important because their solutions have a special property: if \(y_1(x)\) and \(y_2(x)\) are solutions then so is \(cy_1+cy_2\) for any constant solution for any constants \(c_1,c_2\). This means that the solutions to a homogeneous differential equation form a “linear combination” of a set of functions, and this set of functions is a basis for the solution space.

To find the general solution of a second-order homogeneous differential equation, we can use the characteristic polynomial. The characteristic polynomial is obtained by replacing \(y”\) and \(y’\) in the differential equation with the variables \(m^2\) and \(m\), respectively. The resulting polynomial is called the characteristic polynomial and is usually of degree 2.

Solving the characteristic polynomial will give us the roots \(\lambda\) and \(\mu\). Depending on whether these roots are real and distinct, repeated, or complex conjugates, we can then write the general solution of the differential equation using a combination of exponential functions or trigonometric functions.

Overall, homogeneous differential equations are a fundamental concept in differential equations and have important applications in many areas of science and engineering, including physics, chemistry, and economics.