To solve the differential equation [tex]\( 9 y'' + \pi^2 y = 0 \)[/tex], we follow these steps:
1. Set up the equation:
The differential equation given is:
[tex]\[
9 y'' + \pi^2 y = 0
\][/tex]
2. Transform the differential equation:
Divide through by 9 to obtain a simpler form:
[tex]\[
y'' + \frac{\pi^2}{9} y = 0
\][/tex]
3. Identify the characteristic equation:
This is a second-order linear homogeneous differential equation with constant coefficients. The general form is:
[tex]\[
y'' + \lambda y = 0
\][/tex]
Comparing, we have [tex]\(\lambda = \frac{\pi^2}{9}\)[/tex].
4. Solve the characteristic equation:
The characteristic equation corresponding to the differential equation is:
[tex]\[
r^2 + \frac{\pi^2}{9} = 0
\][/tex]
Solving for [tex]\(r\)[/tex], we get:
[tex]\[
r^2 = -\frac{\pi^2}{9}
\][/tex]
Taking the square root of both sides, we obtain:
[tex]\[
r = \pm \frac{i \pi}{3}
\][/tex]
Here, [tex]\(i\)[/tex] is the imaginary unit.
5. Write the general solution:
Because the roots [tex]\(r = \pm \frac{i \pi}{3}\)[/tex] are purely imaginary, the general solution to the differential equation is of the form:
[tex]\[
y(x) = C_1 \cos\left(\frac{\pi x}{3}\right) + C_2 \sin\left(\frac{\pi x}{3}\right)
\][/tex]
where [tex]\(C_1\)[/tex] and [tex]\(C_2\)[/tex] are arbitrary constants.
6. Final answer:
Thus, the solution to the differential equation [tex]\( 9 y'' + \pi^2 y = 0 \)[/tex] is:
[tex]\[
y(x) = C_1 \sin\left(\frac{\pi x}{3}\right) + C_2 \cos\left(\frac{\pi x}{3}\right)
\][/tex]
This completes the detailed step-by-step solution to the differential equation.
