BasicTheoryLinear

BasicTheoryLinear - Basic Theory of Linear O.D.E.s...

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Basic Theory of Linear O.D.E.’s Definition : A linear O.D.E. of order n, in the independent variable x and the dependent variable y is an equation that can be expressed in the form: a 0 (x)y n ( ) + a 1 (x)y n ! 1 ( ) + ..... + a n ! 1 (x) " y + a n (x)y = b(x) We will assume that a 0 (x), a 1 (x), …. , a n (x) and b(x) are continuous real-valued functions on the interval [a, b] and a 0 (x) 0 for at least one x in [a, b]. Remark : A linear ODE satisfies the following conditions: 1- The dependent variable y and its derivatives occur to the first degree only. 2- No products of y and/or any of its derivatives appear in the equation. 3- No transcendental functions of y and/or its derivatives occur. Definition : Equations that not satisfy the above conditions are called nonlinear. Remark : 1) The functions a 0 (x), a 1 (x), …. , a n (x) are called the coefficients of the equation. 2) The right-hand member b(x) is called the non-homogeneous term. 3) If b(x) = 0, then the equation reduces to a 0 (x)y n ( ) + a 1 (x)y n ! 1 ( ) + ..... + a n ! 1 (x) " y + a n (x)y = 0 and it is called a homogeneous linear differential equation (H.L.D.E.) . Example : 1) d 3 y dx 3 ! cos(x) d 2 y dx 2 + 3x dy dx + x 3 y = e x is a linear differential equation of order 3, where a 0 (x) = 1, a 2 (x) = - cos(x), a 3 (x) = 3x, a 4 (x) = x 3 and b(x) = e x , they are all continuous function on ! . Definition : If a 0 , a 1 , …. , a n are all constants, then we have a linear differential equation with constant coefficients, otherwise we have an equation with variable coefficients. Example: 1) 3y’’’ + 2y’’ – 4y’ – 7y = sin x is an equation with constant coefficients. 2) 2y’’ + xy’ – e x y = 0 is an equation with variable coefficients. Theorem : Let a 0 (x)y n ( ) + a 1 (x)y n ! 1 ( ) + ..... + a n ! 1 (x) " y + a n (x)y = b(x) with a 0 (x), a 1 (x), …. , a n (x) and b(x) are continuous real-valued functions on the interval [a, b] and a 0 (x) 0 for at least one x in [a, b]. Let x 0 be a point on the interval [a, b] and let c 0 , c 1 , c n-1 be n arbitrary constants. Conclusion : There exists a unique function f(x) defined on [a, b] such that f(x) is a solution of the linear differential equation and satisfies f(x 0 ) = c 0 , f’(x 0 ) = c 1 , f’’(x 0 ) = c 2 , … , f (n-1) (x 0 ) = c n-1 .
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Remark : The theorem extends the I.V.P. discussed for first order differential equations to linear differential equations of order n. Example : Consider the I.V.P. d 2 y dx 2 + 3x dy dx + x 3 y = e x y(1) = 2 y'(1) = ! 5 " # $ $ % $ $ a 0 (x) = 1, a 1 (x) = 3x, a 3 (x) = x 3 , and b(x) = e x , they are continuous functions everywhere. We have a linear differential equation order 2. The real numbers c 0 = 2 and c 1 = -5. Then, according to the theorem there exists an unique function f(x) continuous
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BasicTheoryLinear - Basic Theory of Linear O.D.E.s...

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