Using characteristic equations is important when working with homogeneous differential equations. Knowing how to find characteristic equations and knowing how to utilize them are prerequisites if you want to know linear homogeneous differential equations by heart.
The characteristic equations are essential when solving linear homogeneous differential equations. We use the roots of the characteristic equation to establish the general solution of the homogeneous differential equation.
Our discussion’s goal is to make sure that you know how to find the characteristic equations from linear homogeneous differential equations. We’ll also show you the significance of the characteristic equations when solving differential equations and have provided some examples for you to work on as well.
The characteristic equation of a linear and homogeneous differential equation is an algebraic equation we use to solve these types of equations. Here’s an example of a pair of a homogeneous differential equation and its corresponding characteristic equation:
Now, let’s generalize this for all second order linear homogeneous differential equations with a general form, as shown below.
The easiest solution we can try out for this equation is $y = e^$, so we’ll also have $y^ <\prime>= re^$ and $y^ <\prime\prime>= r^2e^$. Substitute these expressions into the differential equation.
Since $e^$ will never be equal to zero, the next factor, $ ar^2+ br + c$, must be equal to zero for $y = e^$ to be a zero.
\begin ar^2+ br + c &= 0\end
We can solve for $r$ using algebraic techniques we’ve learned in the past. These roots will determine the form of the homogeneous solution’s nature. The process for establishing rules for characteristic equations for higher order and more complex homogeneous equations will be similar.
From the previous section, we find the characteristic equation by assigning the differentials as a variable. For example, when working with second order homogeneous equations, $ay^ <\prime\prime>+ by^ <\prime>+ cy =0$.
We’ve shown that we can rewrite the following: $y^<\prime \prime>$ with $r^2$, $y^<\prime>$ with $r$, and $y$ with $1$. Here are a few more examples of linear homogeneous differential equations and their corresponding characteristic equations:
Linear Homogeneous Differential Equations
Characteristic Equations