University:
College
Course:
College Algebra
Academic year:

2023
Views:

12

Pages:

7
Author:

Ramy M.

Second order differential equations In this chapter we will be looking exclusively at linear second order differential equations (1) where When when are constants , we call the differential equation homogeneous and we call the differential equation nonhomogeneous. The general solution equals the sum of the homogeneous solution and the particular solution. I - The homogeneous solution (2) Let is a solution of equation (2), then and since This equation is typically called the characteristic equation. This characteristic equation is quadratic and so will have two roots . The roots will have three possible forms. These are 1. Real and distinct roots The general solution is and 2. Real and repeated roots, The general solution is 3. Complex root The general solution is Example 1 Solve the following differential equations 1Solution The characteristic equation is real & distinct 2Solution The characteristic equation is real& repeat 3Solution The characteristic equation is complex 4Solution The characteristic equation is √ √ Try to solve Solve the following differential equations 12345- II- The Particular solution In this section we will take a look at the first method that can be used to find a particular solution to a nonhomogeneous differential equation . 1- IF Example 2 Solve the following differential equations Solution The characteristic equation is real & distinct Suppose the particular solution is Then, compare with if there is a similar term, multiply Get the first and second derivatives = Plugging into the differential equation given by From equations 1& 2 2- IF Example 3 Solve the following differential equations Solution The characteristic equation is real & distinct Suppose the particular solution is Get the first and second derivatives Plugging into the differential equation given Compare the coefficient of Compare the coefficient of from equations 1& 2 3- IF or Example 4 Solve the following differential equations Solution The characteristic equation is real & distinct Suppose the particular solution is Get the first and second derivatives Plugging into the differential equation given Compare the coefficient of Compare the coefficient of from equations 1& 2 Try to solve Solve the following differential equations 1. 2. 3. 4. 5.