A differential equation is an equation that includes a derivative. First Order Ordinary Differential Equations in one variable (x) can be solved in a straightforward fashion simply by taking the antiderivative of the equation statement. The differential equation itself, dy/dx = f(x), says "the slope of the graph of y is equal to the value of f(x) at each x". Recall that there are an infinite number of antiderivatives because of the constant of integration C. Thus, the general solution of dy/dx = f(x) is y = F(x) + C, where F'(x) = f(x). Since adding a constant C to a function is equivalent to shifting its graph vertically, the collection of General Solutions to the differential equation are identical "parallel" graphs, graphs shifted vertically by all possible values of C. But, if we specify that the solution must pass through a point (x0, y0), we nail the solution down to a particular value of C. This solution is called the Particular Solution to the differential equation. Rearranging the general solution we have C = y - F(x), and substituting the initial condition (x0, y0) we have C = y0 - F(x0).
Enter an expression in x in the "dy/dx =" box. Drag the red point to a coordinate representing the initial condition (x0, y0). Notice that a value of C is calculated along with the antiderivative to plot the graph of the Particular Solution that satisfies the Initial Condition. If you check the box "Show General Sol'n", graphs are displayed of the General Solution with values of C from -20 to +20 (in steps of 1).