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

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.