Elementary Second-Order Differential Equations

An elementary second-order differential equation takes the form: $$ y'' = f(x) $$

To determine the general solution, one simply integrates the second derivative twice.

This process yields the first derivative \( y' \), followed by the original function \( y \).

$$ y' = \int y'' \ dx $$

$$ y = \int y' \ dx $$

Equations of this type are among the simplest second-order differential equations to solve.

A Practical Example

Let’s examine the following second-order differential equation:

$$ y'' = \sin x $$

To solve it, we carry out two successive integrations.

First, integrating the second derivative gives the first derivative: \( y' = -\cos x + c_1 \)

$$ \int y'' = \int \sin x \ dx $$

$$ y' = -\cos x + c_1 $$

Next, integrating \( y' \) yields the original function \( y \): \( y = -\sin x + c_1 x + c_2 \)

$$ \int y' = \int (-\cos x + c_1) \ dx $$

$$ y = -\sin x + c_1 \cdot x + c_2 $$

This is the general solution of the differential equation.

And so forth.