x̄ - > Differential equation

 

Differential equations describe the relationship between a function and its derivatives and are widely used in physics, engineering, and biology. They come in different types, depending on the order of the equation and the type of function involved. 

First-order differential equations involve the first derivative of a function and are commonly used to model physical systems that involve rates of change. Second-order differential equations involve the second derivative of a function and are used to model systems that involve acceleration and oscillations. Partial differential equations involve partial derivatives of a function and are useful in modeling systems that involve multiple variables, such as heat transfer and fluid dynamics.

To illustrate how differential equations are solved, let's consider the first-order differential equation:

```

dy/dx = -2xy

```

This equation describes the rate of change of y in relation to the product of x and y. By separating the variables and integrating both sides, we obtain the general solution:

```

dy/y = -2x dx

ln|y| = -x^2 + C

|y| = e^(-x^2 + C)

y = ± e^(-x^2 + C)

```

Here, C is a constant of integration. To find the particular solution that satisfies the initial condition y(0) = 1, we substitute x = 0 and y = 1:

```

1 = ± e^(C)

C = 0

y = e^(-x^2)

```

Thus, the particular solution that satisfies the initial condition is y = e^(-x^2).

In summary, differential equations are a powerful tool for modeling physical systems and understanding the relationship between variables and their derivatives. By solving differential equations, we can obtain specific solutions that help us better understand the behavior of a system.

y''(x) + y(x) = 0

y''(x) = -y(x)

y''(x) + y(x) = 0

second-order linear ordinary differential equation

y''(x) = -y(x)

y(x) = c_2 sin(x) + c_1 cos(x)