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.
