ODE - Chapter 1 Ordinary Differential Equation Page 1...

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Chapter 1 Ordinary Differential Equation Page 1 1 Chapter 1 Ordinary Differential Equation 1.1 Differential Equation Equation, which involves the solving of an unknown variable or function, is a very important topic in mathematics and physics. For example, one of the very simple equation is quadratic equation, for which the variable is to be solved, like and the solution is . Likewise, as you will see later, a differential equation is also an equation, for which the solution of a function is to be solved. Some definition of differential equation A differential equation is a equation involving a function that is going to be solved, for example y(x) and some combinations of the differentials of y(x) and any other functions. For example: (i) (ii) , etc. The degree of an differential equation: The order of a d.e. is the order of the highest derivative appeared in the d.e. For example for eq.(i) and eq(ii), the orders are 3 and 1 respectively. Linear and Non-linear d.e. A linear d.e . is a d.e. that can be written as : (1) where a 1 (x), . ...a n-1 (x) and f(x) are function of x. y(x) is the function that is to be solved. Any d.e. that is not linear are non-linear. Homogeneous and Nonhomogeneous d.e. A homo. d.e. is a d.e. such that f(x)=0 and a nonhomo. d.e. is one such that f(x) 0. Solution The solution to a d.e. is the function that satisfies the equation. However, it is easily found that a d.e. generally have infinite soultions, for example, considering: and the solution is y(x)=4x+C , where C is any constants. In this case the solution is called the general solution of the d.e. The particular solution is a solution that is one of the general solution, for example for the previous case, y(x)=4x+1, y(x)=4x-2 and etc. are particular solutions.
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Chapter 1 Ordinary Differential Equation Page 2 2 However, in some case, a definite solution can possibly be obtained if some more conditions, namely the boundary condition and the initial condition, are given. Example 1 , where C and D are some constants y(x)=5x 2 +Cx+D is the general solution. If some more conditions are given as : y(0)=0 and y’(0)=1 then, a definite solution of y can be found as C=1 and D=0, ie. y(x)=5x 2 +x . Initial value problem: The further conditions are given at the same value of the variable x and different order of derivative of the function y : y(x 0 )=b 0 , y’(x 0 )=b 1 , etc. Boundary value problem: The further conditions are given as the function value y at different values of the variable x , ie. y(x 0 )=b 0 , y(x 1 )=b 1 etc. Consider the d.e. of example 1, y(0)=0 and y’(0)=1 is an initial value problem and y(0)=1 and y(2)=3 is a boundary value problem. As a matter of fact, d.e. is very frequently seen in physical problem and we will illustrate two trivial kinds of examples as below. Object fallen under gravity
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This note was uploaded on 08/17/2011 for the course PHYS 230 taught by Professor Harris during the Winter '07 term at McGill.

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ODE - Chapter 1 Ordinary Differential Equation Page 1...

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