Higher Order ODE's (3A)
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The Properties of a Line Equation (1) y = x f (1) = 1 f (2) = f (1+1)=f (1) + f (1) = 2 f (2) = 2 f (3) = f (2+1)=f (2) + f (1) = 3 f (3) = 3 f (4) = f (3+1)=f (3) + f (1) = 4 y = x f (0.1) = 0.1 f (0.1) = f (0.1 1)=0.1 f (1) = 0.1 f (0.5) = 0.5 f (0.5) = f (0.5 1)=0.5 f (1) = 0.5 f (1.5) = 1.5 f (1.5) = f (1.5 1)=1.5 f (1) = 1.5 Higher Order ODEs (3A) 3
The Properties of a Line Equation (2) y = x f (1) = 1 f (2) = 2 f (3) = 3 f (4) = 4 f (0.5) = 0.5 f (1.0) = 1.0 f (1.5) = 1.5 f (2.0) = 2.0 f (0.1) = 0.1 f (0.2) = 0.2 f (0.3) = 0.3 f (0.4) = 0.4 y = x f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) f (k x) = k f (x) Higher Order ODEs (3A) 4
Linearity Property f ( x 1 + x 2 ) f ( x 2 ) f ( x 1 ) y = a x Additivity f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) a (x 1 + x 2 ) = a (x 1 ) + a ( x 2 ) x 1 x 2 x 1 +x 2 y = a x k f (x) f ( x) Homogeneity f (k x) = k f ( x) x k x a (k x) = k(a x) Higher Order ODEs (3A) 5
Linearity & Affinity Linearity Affinity y = a x y = a x + b Translation from linear + -ity. from the Latin, affinis, "connected with" f (x 1 + x 2 ) = f (x 1 ) + f ( x 2 ) f (k x) = k f (x) Additivity Homogeneity Additivity Homogeneity Higher Order ODEs (3A) 6
Linear Map Additivity f (x 1 + x 2 ) = f (x 1 ) + f (x 2 ) x 1 f (x) f (x 1 ) x 1 + x 2 f (x) f (x 1 ) + f ( x 2 ) x 2 f (x) f (x 2 ) Homogeneity f (k f (k x) = k f (x) f ( x) x f (x) f (x) k x f (x) k f ( x) Higher Order ODEs (3A) 7
Linear Operators Additivity d d f [f + g] = d x d x + d g d x f (x) d d x f ' (x) f (x) + g( x ) d d x f ' (x) + g' (x) g( x) d g' (x) d x Homogeneity d d x [k f ] = k d f d x f (k x) = k f ( x) f (x) d f ' (x) k f ( x) d k f '( x) d x d x Higher Order ODEs (3A) 8
Linear Systems Additivity S{g 1 (x) + g 2 (x)} = S {g 1 ( x)} + S{g 2 ( x)} g 1 (x) S y 1 ( x) g 1 (x) + g 2 ( x) S y 1 ( x) + y 2 ( x) g 2 ( x) S y 2 ( x) d 2 y d x 2 + a 1 d y d x + a 2 y = g( x) Homogeneity f (k x) = k f ( x) S{k f (x)} = k S{f (x)} g( x) S y(x) k g( x) S k y(x) Higher Order ODEs (3A) 9
Differential Operator Differential Operator D y(x) y '( x) D = d dx D y = d y dx N-th Order Differential Operator y(x) L L( y(x)) L = a n ( x) D n + a n 1 ( x) D n 1 + + a 1 ( x)d + a 0 ( x) L( y) = a n ( x) dn y d x n + a n 1(x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0( x) y Higher Order ODEs (3A) 10
Examples f (x) d d x f ' (x) d f d x = f '(x) D (D f ) = d dx ( d f d x ) = f ' ' (x) f (x) D f ' (x) D f = f '( x) D 2 f = d2 f d x 2 = f ' ' (x) Differential Operator : Linear D (c f (x)) = c D f (x) D (f (x) + g(x)) = D f ( x) + D g(x) D(α f ( x) + β g( x)) = α D f ( x) + β D g(x) n-th order Differential Operator L = a n (x)d n + a n 1 (x)d n 1 + + a 1 ( x) D + a 0 (x) (D 2 + 2 D + 1) f (x) = D 2 f ( x) + 2 D f ( x) + f (x) = f ' '( x) + 2 f '(x) + f ( x) n-th order Differential Equations using the Differential Operator y' ' + 5 y '+2 y = 3 x D 2 + 5 D + 2 = L L( y) = 0 L( y) = g(x) y' ' + 5 y '+2 y = 0 y' ' + 5 y '+2 y = 3 x Higher Order ODEs (3A) 11
Linear System Linear System g( x) S y(x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) (a n (x) D n + a n 1 (x) D n 1 + + a 1 ( x) D + a 0 (x))( y( x)) = g( x) L( y(x)) = g(x) Higher Order ODEs (3A) 12
Linear Differential Equations a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = 0 Non-homogeneous Equation Homogeneous Equation Higher Order ODEs (3A) 13
Linear Differential Equations Non-homogeneous Equation a n ( x) dn y 1 d x n + a n 1(x) d n 1 y 1 d x + + a 1(x) d y 1 n 1 d x + a 0(x) y 1 = g 1 (x) g 1 (x) y 1 (x) a n ( x) dn y 2 d x n + a n 1(x) d n 1 y 2 d x + + a 1(x) d y 2 n 1 d x + a 0(x) y 2 = g 2 (x) g 2 (x) y 2 (x) a n ( x) dn y d x n + a n 1(x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0(x) y = g(x) g 1 +g 2 y 1 + y 2 Superposition Non-homogeneous Equation a n (x) dn k y 1 d x n + a n 1 ( x) d n 1 k y 1 + + a d x n 1 1 ( x) d k y 1 d x + a 0 ( x) k y 1 = k g 1 ( x) g 1 (x) y 1 (x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0( x) y = g(x) k g 1 (x) k y 2 ( x) Higher Order ODEs (3A) 14
Homogeneous Equation a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = 0 Associated Homogeneous Equation d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = g( x) d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = 0 Associated Homogeneous Equation with constant coefficients a n m n + a n 1 m n 1 + + a 1 m + a 0 = 0 Auxiliary Equation m = m 1, m 2,, m n y = c 1 e m 1 x + c 2 e m 2x + + c n e m n x n solutions of the Auxiliary Equation General Solutions of the Homogeneous Equation Higher Order ODEs (3A) 15
Non-homogeneous Equation a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = g(x) a n (x) dn y d x n + a n 1( x) dn 1 y d x n 1 + + a 1( x) d y d x + a 0( x) y = 0 Associated Homogeneous Equation d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = g( x) d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = 0 Associated Homogeneous Equation with constant coefficients Particular Solution y = c 1 e m 1 x + c 2 e m 2 x + + c n e m n x + y p ( x) General Solutions of the Non-homogeneous Equation Higher Order ODEs (3A) 16
Initial Value Problem d n y d x n = f (x, y, y ',, y (n 1) ) on some interval I containing x 0 General Form d n 1 d x n 1 y(x 0) = k n 1 d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 IVP Higher Order ODEs (3A) 17
Initial Value Problem variable coefficients a n ( x) dn y d x n + a n 1(x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0(x) y = g(x) Linear Equation with variable coefficients d n 1 d x n 1 y(x 0) = k n 1 d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 IVP Higher Order ODEs (3A) 18
Initial Value Problem constant coefficients n +1 d n y a n d x + a d n 1 y n n 1 d x + + a d y n 1 1 d x + a 0 y = g(x) Linear Equation with constant coefficients n d n 1 d x n 1 y(x 0) = y (n 1) (x 0 ) = k n 1 n d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 y = c 1 e m 1 x + c 2 e m 2 x + + c n e m n x + y p ( x) n Parameters c i Higher Order ODEs (3A) 19
Boundary Value Problem a 2 (x) d2 y d x 2 + a 1(x) d y d x + a 0(x) y = g(x) y(a) = y 0 y' (a) = y 0 y(a) = y 0 y' (a) = y 0 Various Boundary Conditions y(b) = y 1 y(b) = y 1 y' (b) = y 1 y' (b) = y 1 Higher Order ODEs (3A) 20
Existence of a unique solution (1) a n ( x) dn y d x n + a n 1(x) dn 1 y d x n 1 + + a 1(x) d y d x + a 0(x) y = g(x) Non-homogeneous Equation with variable coefficients d n 1 d x n 1 y(x 0) = k n 1 d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 IVP a n ( x), a n 1 (x), a 1 (x), a 0 ( x), are all continuous on the interval I and a n ( x) 0 g(x) The solution y(x) of the IVP 1) exists on the interval I 2) is unique Higher Order ODEs (3A) 21
Existence of a unique solution (2) d y d x = f (x, y) on some interval I containing x 0 d n 1 d x n 1 y(x 0) = k n 1 d d x y(x 0) = y' (x 0 ) = k 1 n Initial Conditions at x = x 0 y(x 0 ) = y(x 0 ) = k 0 IVP f (x, y) And f x are continuous on R The solution y(x) of the IVP 1) exists on the interval I0 2) is unique R a x b I0 x 0 h x x 0 +h (h>0) c y d contained in [a, b] Higher Order ODEs (3A) 22
Continuous Function a continuous function is a function for which, intuitively, "small" changes in the input result in "small" changes in the output. Otherwise, a function is said to be a "discontinuous function". A continuous function with a continuous inverse function is called a homeomorphism. Removable discontinuity Jump discontinuity Essential discontinuity Higher Order ODEs (3A) 23
Differentiable Function a differentiable function of one real variable is a function whose derivative exists at each point in its domain. the graph of a differentiable function must have a non-vertical tangent line at each point in its domain, be relatively smooth, and cannot contain any breaks, bends, or cusps. not differentiable at x=0 Higher Order ODEs (3A) 24 a differentiable function
Differentiability and Continuity If f is differentiable at a point x0, then f must also be continuous at x0. any differentiable function must be continuous at every point in its domain. The converse does not hold: a continuous function need not be differentiable. Differentiable Continuous not differentiable at x=0 Higher Order ODEs (3A) 25
Check for Linear Independent Solutions Homogeneous Linear n-th order differential equation a n (x) d n y d x + a n 1(x) dn 1 y n d x n 1 + + a 1 (x) d y d x + a 0(x) y = 0 n-th order Homogeneous y 1, y 2,, y n n linearly independent solutions W ( y 1, y 2,, y n ) 0 {y 1, y 2,, y n } fundamental set of solutions y = c 1 y 1 + c 2 y 2 + + c n y n general solution The general solution for a homogeneous linear n-th order differential equation Higher Order ODEs (3A) 26
References [1] http://en.wikipedia.org/ [2] M.L. Boas, Mathematical Methods in the Physical Sciences [3] E. Kreyszig, Advanced Engineering Mathematics [4] D. G. Zill, W. S. Wright, Advanced Engineering Mathematics [5] www.chem.arizona.edu/~salzmanr/480a