Question

Answer Problems 1-10 without referring back to the text. Fill in the blank or answer true or false.

If $y=1-x+6x^2+3e^x$is a solution of a homogeneous fourth-order linear differential equation with constant coefficients, then the roots of the auxiliary equation are_____

Short answer

m₁= 1 & m₂= 0 with multiplicity of 3

Step-by-step solution

Definition

Auxiliary equation is an equation obtained from the standard form of a linear differential equation by replacing the right member by zero

Find root

Consider the equation $y=1-x+6x^2+3e^x$---(1)

Since the differential equation is fourth order, we know there must be four roots to the auxiliary equation.

Looking at equation (1), there is a single exponential function which is 3 e^x. Given the exponent of this function,

We know that one root is m₁ = 1

Find root

Looking at the first three terms, we have.

$1-x+6x^2$

Notice that they are all power functions without any exponential function.

This means that the root must be m₂= 0.

Since the exponential function is of the form e^0x.

Moreover, this root is has a multiplicity of three since there are three terms that are power functions. So, we have the roots of the auxiliary equation: m₁= 1 & m₂= 0 with multiplicity of 3