General and Particular Solutions
Differential equations appear in almost every area of daily life including science, business, and many others. We will only consider ordinary differential equations (ODE). An ODE is a relation on a function of one independent variable and the derivatives of with respect to , i.e. . For example, .
An ODE is linear if can be written as a linear combination of the derivatives of , i.e. . A linear ODE is homogeneous if .
A general solution to a linear ODE is a solution containing a number (the order of the ODE) of arbitrary variables corresponding to the constants of integration. A particular solution is derived from the general solution by setting the constants to particular values. For example, for linear ODE of second degree , a general solution has the form where are real numbers. By setting and
It is generally hard to find the solution of differential equations. Graphically and numerical methods are often used. In some cases, analytical method works, and in the best case, has an explicit formula in .
For a video introduction to differential equations (27.0), see Math Video Tutorials by James Sousa, Introduction to Differential Equations (8:12).
Slope Fields and Isoclines
We now only consider linear ODE of the first degree, i.e. . In general, the solutions of a differential equation could be visualized before trying an analytic method. A solution curve is the curve that represents a solution (in the plane).
The slope field of the differential |eq|uation is the set of all short line segments through each point and with slope .
An isocline (for constant ) is the line along which the solution curves have the same gradient . By calculating this gradient for each isocline, the slope field can be visualized; making it relatively easy to sketch approxi- mate solution curves. For example,. The isoclines are .
Example 1 Consider . We briefly sketch the slope field as above.
The solutions are .
- Sketch the slope field of the differential equation . Sketch the solution curves based on it.
- Sketch the slope field of the differential equation . Find the isoclines and sketch a solution curve that passes through .
Differential Equations and Integration
We begin the analytic solutions of differential equations with a simple type where is a function of only. is a function of . Then any antiderivative of is a solution by the Fundamental Theorem of Calculus:
Example 1 Solve the differential equation with .
Solution. . Then gives , i.e. Therefore .
- Solve the differential equation with .
Solving Separable First-Order Differential Equations
The next type of differential equation where analytic solution are relatively easy is when the dependence of on and are separable: where is the product of a functions of and respectively. The solution is in the form . Here is never or the values of in the solutions will be restricted by where .
Example 1 Solve the differential equation with the initial condition .
Solution. Separating and turns the equation in differential form . Integrating both sides, we have .
Then gives , i.e. and .
Therefore, the solutions are .
Here is when and the values of in the solutions satisfy or .
Example 2. Solve the differential equation .
Solution. Separating and turns the equation in differential form
Resolving the partial fraction gives linear equations and .
So . Integrating both sides, we have or with . Then , i.e. where .
Therefore, the solution has form where .
- Solve the differential equation which satisfies the condition .
- Solve the differential equation .
- Solve the differential equation .
Exponential and Logistic Growth
In some models, the population grows at a rate proportional to the current population without restrictions. The population is given by the differential equation , where is the growth rate. In a refined model, the rate of growth is adjusted by another factor where is the carrier capacity. This is close to when is small compared with but close to when is close to .
Both differential equations are separable and could be solved as in last section. . The solutions are respectively:
and with .
Example 1 (Exponential Growth) The population of a group of immigrants increased from to from the end of the first year to the end of second year they came to an island. Assuming an exponential growth model on the population, estimate the size of the group of initial immigrants.
Solution. The population of the group is given by where the initial population and relative growth rate are to be determined.
At (year), , so .
At (year), , so .
Dividing both sides of the second equation by the first, we have .
Then back in the first equation, . So . There are initial immigrants.
Example 2 (Logistic Growth) The population on an island is given by the equation . Find the population sizes . At what time will the population first exceed ?
Solution. The solution is given by where .
Solve for time, gives . So . The population first exceed in the year.
- (Exponential Growth) The population of a suburban city increased from in 2005 to in 2007. Assuming an exponential growth model on the population, by which year will the population first exceed ?
- (Logistic Growth) The population of a city is given by the equation . Find the population sizes . At what time will the population first exceed ?
For a video presentation of Differential Equations including growth and decay (27.0), see Differential Equations, Growth and Decay (7:23).
Numerical Methods (Euler's, Improved Euler, Runge-Kutta)
The Euler's method is a numerical approximation to a solution curve starting from the point through the algorithm:
where and is the step size.
The shorter step size, the better is the approximation to the solution curve.
Improved Euler (Heun) method adapts on Euler's method by using both end point values:
Since also appears on the right side, we replace it by Euler's formula,
The Runge-Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of our ODE. On them, apply Simpson's rule:
Exercise 1. Apply the Euler's, improved Euler's and the Runge-Kutta methods on the ODE
to approximate the solution that satisfy from to with .
We know the exact solution is . Compare their relative accuracy against the exact solution.
Texas Instruments Resources
In the CK-12 Texas Instruments Calculus FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9732.