# Solve the second-order differential equation $$\frac{d^2y}{dx^2} + y = 0$$

### Solve the second-order differential equation $$\frac{d^2y}{dx^2} + y = 0$$

A: $$y = C_1 \cos(x) + C_2 \sin(x)$$

### Worksheet

Step 1: Recognize the Form

This is a linear homogeneous second-order differential equation with constant coefficients. The general form of such an equation is:
$a \frac{d^2y}{dx^2} + b \frac{dy}{dx} + cy = 0$
For our equation, $$a = 1$$, $$b = 0$$, and $$c = 1$$. So, we have:
$\frac{d^2y}{dx^2} + y = 0$

Step 2: Write the Characteristic Equation

To solve this differential equation, we first write the characteristic equation. Assume a solution of the form $$y = e^{rx}$$. Substitute $$y = e^{rx}$$ into the differential equation:
$\frac{d^2}{dx^2}(e^{rx}) + e^{rx} = 0$
The second derivative of $$y = e^{rx}$$ is $$r^2 e^{rx}$$. So, we get:
$r^2 e^{rx} + e^{rx} = 0$
$e^{rx} (r^2 + 1) = 0$

Since $$e^{rx}$$ is never zero, we can divide both sides by $$e^{rx}$$:
$r^2 + 1 = 0$

Step 3: Solve the Characteristic Equation

Solve the characteristic equation for $$r$$:
$r^2 + 1 = 0$
$r^2 = -1$
$r = \pm i$

Step 4: Write the General Solution

The roots of the characteristic equation are $$r = i$$ and $$r = -i$$. For complex roots $$r = \alpha \pm i\beta$$, the general solution to the differential equation is:
$y = e^{\alpha x} (C_1 \cos(\beta x) + C_2 \sin(\beta x))$

In this case, $$\alpha = 0$$ and $$\beta = 1$$, so the solution simplifies to:
$y = C_1 \cos(x) + C_2 \sin(x)$

Final Solution

The general solution to the differential equation $$\frac{d^2y}{dx^2} + y = 0$$ is:
$y = C_1 \cos(x) + C_2 \sin(x)$

where $$C_1$$ and $$C_2$$ are arbitrary constants determined by initial conditions or boundary conditions.

#### Recap and Tips

1. Recognize the Type: Identify the differential equation as a second-order linear homogeneous equation with constant coefficients.
2. Characteristic Equation: Formulate the characteristic equation by assuming a solution of the form $$y = e^{rx}$$.
3. Solve for Roots: Solve the characteristic equation for $$r$$.
4. General Solution: Write the general solution based on the roots. For real roots, it's a combination of exponentials; for complex roots, it's a combination of sines and cosines.