Question

Answer Problems 1-10 without referring back to the text. Fill in the blank or answer true or false. If $\mathrm{y}={\mathrm{c}}_1{\mathrm{x}}^2+{\mathrm{c}}_2{\mathrm{x}}^{2}In\mathrm{x},\mathrm{x}>0$, is the general solution of a homogeneous secondorder Cauchy-Euler equation, then the DE is_

Short answer

$x^2\frac{d^{2}y}{d{x}^2}-3x\frac{dy}{dx}+4y=0$

Step-by-step solution

Definition

A differential equation is an equation with one or more derivatives of a function

Find root

The given solution is

For Cauchy-Euler equations, auxiliary equations that have repeated roots lead to the general solution of $y=c_1{x}^{m_1}+c_2{x}^{m_2}\mathrm{In}x$

This is the exact form of the solution given in question. This means that the root of the auxiliary equation is a repeated root, and it is equal to m₁ = 2.

Form auxiliary equation

Using this root, the auxiliary equation is

$$\begin{gathered} {\left(m-2\right)}^2=0 \\ {m}^2-4m+4=0 \end{gathered}$$

In general, the auxiliary equation takes the form of $a{m}^2+(b-a)m+c=0$

Comparing the equation above to the auxiliary equation for our problem

a = 1, c = 4 and b is given by:

b - 1 = - 4

b = -3

Find equation

Using the constants we found, we can now write out the Cauchy-Euler equation.

In general, the Cauchy-Euler equation takes the form of $a{x}^2\frac{d^{2}y}{d{x}^2}+bx\frac{dy}{dx}+cy=0$

Substituting the constants in the above equation we have:

$x^2\frac{d^{2}y}{d{x}^2}-3x\frac{dy}{dx}+4y=0$