Direct integration

Differential equations where the function representing the rate of change is known in terms of the independent variable only are the simplest kind. The function which satisfies the differential equation is its solution. The dependent variable is the variable which represents the solution, and the independent variable is the variable which the solution depends on.

For the equation

$$\frac{\mathrm{d}y}{\mathrm{d}x} = f'(x)$$

the independent variable is and the dependent variable is . The solution to this equation can be found by simply integrating both sides to give

$$\begin{eqnarray*} \int \frac{\mathrm{d}y}{\mathrm{d}x} dx & = & \int f'(x) dx \\ y & = & f(x) + c \end{eqnarray*}$$

This form of the solution, where the value of the constant (or constants, for higher order equations) is not known is called the general solution. The solution whose curve passes through the points , say, is called the particular solution, where the value of the constant is known. Given the point, the value of the constant can be found by substituting the coordinates of the point into the equation and solving it to find the constant.

As an example, consider the equation . To solve it, we integrate both sides to give . If this curve passes through the point , we can deduce the value of the constant by substituting into the equation as follows.

$$\begin{eqnarray*} 0 & = & 0 - \cos{0} + c \\ 0 & = & - 1 + c \\ c & = & 1 \end{eqnarray*}$$

This gives

$$y = x^2 - \cos{x} + 1$$