Introduction to Differential Equations Second order Linear Differential Equations - Mathematics Assignment Help

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Introduction Differential equations are equations that involve derivatives. They arise in mathematical modelling whenever we have information about the rate of change of some quantity. You have met some of these already . Preliminary Example: A car is stationary at an intersection. It accelerates from rest when the driver notices the traffic light turn green. Initially the acceleration is 10 m/s 2 , but it decreases linearly and is zero after 8 s. Find the velocity of the car during this time interval .

Note that this example involves only the first derivative (it is called a first order DE), and its solution involves one arbitrary constant. We will call this the general solution. We have information that enables us to find the value of that constant . MM1 – Introduction to Differential Equations 2 Note that solving the differential equation gives us a function. We may know an initial value for the function that solves the DE. In this case we say we have a DE with an initial value, and together this makes up an Initial Value Problem (IVP). In the previous example, we solved the IVP ???????? ???????? = 10 − 5 4 ????, ???? 0 = 0. The solution was…. MM1 – Introduction to Differential Equations 3 Example – Second Order DE Second order DE s involve the second derivative (and may also include the first derivative), and their general solutions usually contain TWO arbitrary constants. These arise in the study of oscillations and vibrations. Later we will learn how to solve these, that is, how to find the solution.

In this example we are given a solution and asked to verify that it works. Verify that the function ???? = ???????? 3???? + ???????? −???? is a solution to the DE ???? ′′ − 2???? ′ − 3???? = 0. MM1 – Introduction to Differential Equations 4 Number of arbitrary constants and definition of general solutions.

Note that the first order DE of our examples had one arbitrary constant in its solution and the second order DE had two arbitrary constants. This pattern will be the case for the DEs in this subject: general solutions of ????th order DEs will have ???? arbitrary constants. However you should be aware that there are cases where this rule does not hold. For example, the second order DE ???? 2???? ???????? 2 = ???????? ???????? 2 has a solution ???? = ???? − log ???? + ???? , but is this a general solution? The DE also has the solution ???? = 1, which is not obtainable by choosing values for ???? and ????. Textbooks may differ in how they define general solutions. For all of the DEs that we will be working with this semester, a general solution to a differential equation of order ???? will be a function with ???? arbitrary constants, which gives all the solutions. MM1 – Introduction to Diffe

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