Differential Equations

In this unit, we will define a differential equation and investigate methods for finding solutions to differential equations. We will then use a slope field to represent the solution of a first order differential equation.

Verify the solution of an ordinary differential equation

Find the general solution and a particular solution of a differential equation

Solve a separable first-order differential equation

A function y = f(x) is called a solution to a differential equation if, when y and its derivatives are substituted into the equation, the equation is satisfied for all x for which f and its derivatives are defined.

The general solution of a differential equation represents all the solutions to the differential equation and is in the form of an equation with arbitrary constants. The number of arbitrary constants in the general solution agrees with the order of the differential equation. General solutions will end in + C.

Any solution of a differential equation obtained from the general solution by assigning values to the arbitrary constants is called a particular solution. A particular solution will be a solution based in the initial condition. An initial condition is a restriction on the variables of the general solution that leads to a particular solution.

If a first order differential equation is said to be separable, then you will be able to do the following with that differential equation to find the original equation that the differential equation came from.

Once you have integrated the differential equation, you will solve it for y in order to get the general solution. If the problem is restricted to specific points, then solve the general equation for c and re-write the answer in terms of a particular solution.

The first part of unit 7 dealt with finding the general and particular solutions using calculus and algebra. In most instances, find the general solution of a first order differential equation is not possible. In these cases, another method that will be used is slope fields.

Using the differential equations, if we draw small line segments representing the tangent lines at various points in the domain of f, the resulting graph is a slope field. This will give a picture of the family of curves that make up the solution to the differential equation. Once you draw the family of curves, you can use an initial condition to draw the graph of the original function through that point.

When drawing a slope field on the exam, be sure to do the following:

• The line segments drawn at the points where the derivative is zero are horizontal

• The line segments drawn at the points where the derivative is undefined are left blank

• The line segments at the points where the derivative is positive is rising from the left to the right and closely represents the slope at that point

• The line segments at the points where the derivative is negative is falling from the left to the right and closely represents the slope at that point

• Be sure each line segment is drawn through the chosen point