Answer :

To solve the differential equation [tex]\(\frac{d^2 y}{d x^2} - 9y = 0\)[/tex], we can follow these systematic steps:

1. Rewrite the Differential Equation:
We start by rewriting the given differential equation in the form:
[tex]\[ y'' - 9y = 0 \][/tex]
where [tex]\(y''\)[/tex] denotes the second derivative of [tex]\(y\)[/tex] with respect to [tex]\(x\)[/tex].

2. Find the Characteristic Equation:
For a differential equation of the form [tex]\(ay'' + by' + cy = 0\)[/tex], we assume a solution of the form [tex]\(y = e^{mx}\)[/tex]. Substituting [tex]\(y = e^{mx}\)[/tex] into the differential equation gives us the characteristic equation. For our equation [tex]\(y'' - 9y = 0\)[/tex]:

[tex]\[ m^2 e^{mx} - 9 e^{mx} = 0 \][/tex]

This simplifies to:

[tex]\[ e^{mx}(m^2 - 9) = 0 \][/tex]

Since [tex]\(e^{mx} \neq 0\)[/tex], we can divide through by [tex]\(e^{mx}\)[/tex]:

[tex]\[ m^2 - 9 = 0 \][/tex]

3. Solve the Characteristic Equation:
Solve [tex]\(m^2 - 9 = 0\)[/tex] for [tex]\(m\)[/tex]:

[tex]\[ m^2 = 9 \][/tex]

Taking the square root of both sides, we get:

[tex]\[ m = \pm 3 \][/tex]

Hence, the roots of the characteristic equation are [tex]\(m_1 = 3\)[/tex] and [tex]\(m_2 = -3\)[/tex].

4. Form the General Solution:
With the roots of the characteristic equation, the general solution to the differential equation is given by:

[tex]\[ y(x) = C_1 e^{3x} + C_2 e^{-3x} \][/tex]

where [tex]\(C_1\)[/tex] and [tex]\(C_2\)[/tex] are arbitrary constants determined by initial conditions if provided.

Thus, the general solution to the differential equation [tex]\(\frac{d^2 y}{d x^2} - 9y = 0\)[/tex] is:

[tex]\[ y(x) = C_1 e^{3x} + C_2 e^{-3x} \][/tex]

This completes the solution, showing each step and reasoning behind solving the homogeneous second-order linear differential equation.