Separable Equations (1A)
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Types of First Order ODEs Separable Equations (1A) 3
First & Second Order IVPs (y=f(x)) First Order Initial Value Problem = g(x, y) y ' = g(x, y) y(x 0 ) = y 0 y(x 0 ) = y 0 Second Order Initial Value Problem d 2 y 2 = g(x, y, y ') y ' ' = g(x, y, y ' ) y(x 0 ) = y 0 y ' (x 0 ) = y 1 y(x 0 ) = y 0 y ' (x 0 ) = y 1 Separable Equations (1A) 4
Types of First Order ODEs A General Form of First Order Differential Equations = g(x, y) y ' = g(x, y) Separable Equations = g 1 (x)g 2 ( y) y ' = g 1 (x)g 2 ( y) y = f (x) Linear Equations a 1 (x) + a 0 (x) y = g(x) a 1 (x) y ' + a 0 (x) y = g(x) y = f (x) Exact Equations M (x, y)dx + N (x, y)dy = 0 z x + z y dy=0 z = f (x, y) Separable Equations (1A) 5
Separable First Order ODEs Separable Equations (1A) 6
Separable ODEs A General Form of First Order Differential Equations = g(x, y) y ' = g(x, y) Separable Equations = g (x)g ( y) y ' = g (x)g 1 2 1 2 ( y) y = f (x) 1 g 2 ( y) = g (x) 1 1 g 2 ( y) y ' = g 1(x) p( y) = q(x) p( y) y ' = q (x) Separable Equations (1A) 7
Solving Separable ODEs (1) p( y) = q(x) p( y) y ' = q (x) y = f (x) not a ratio p( y) p( y) = q(x) p( y) y ' = q (x) dx = q(x)dx p( y) y ' dx = q(x)dx dy = df dx dx p( y) dy = q(x) dx p( y) dy = q (x) dx p( y)dy = q(x)dx p ( y) dy = q(x) dx p( y)dy = q(x)dx p( y) dy = q(x) dx p( y)dy includes a constant q(x)dx includes another constant P( y) = Q(x) + C P( y) = Q(x) + C implicit function y Separable Equations (1A) 8
Solving Separable ODEs (2) p( y) = q(x) p( y) y ' = q (x) given a composite function p(y(x)) fin=f(x) P( y) = p( y) dy + c 1 d [ P( y) ]= p ( y) d [ p ( y) dy + c 1 ] dy dx = q(x) d [ p( y) dy + c 1 ] = q(x) p( y)dy = q(x)dx p( y) dy = q(x) dx d dy [ p( y) dy + c 1 ] y ' = q(x) d [ p( y) dy + c 1 ] = q(x) p( y)dy = q(x)dx p( y) dy = q(x) dx d d dx [ P( y) ] dy dx = q(x ) [ P( y) ] = q (x) p( y) = d [ p ( y) dy + c 1 ] P( y) = Q(x) + C P( y) = Q(x) + C Separable Equations (1A) 9
Direction Fields Separable Equations (1A) 10
Plot of F(x,y)=cos(x) f (x) = sin( x) +1 0-1 0 +1 0-1 0 +1 0-1 0 slope F (x, y) slope=+1 F (x, f (x )) = f '( x) (x i, y i ) f '(x i ) +1 +1 +1 0 0 0 0 0 0 f ' (x) = cos( x) -1-1 -1 Separable Equations (1A) 11
A function of two variables y = f (x) y = f (x) f maps x to y x y u = F (x, y) u = F (x, y) F maps (x,y) to u (x, y) u u = F (x, y) (x, y) u = f '(x) u = F (x, y) F maps (x,y) to f'(x) the derivative of f(x) at x The slope of the tangent line at (x, f(x)) Separable Equations (1A) 12
Direction Field u = F (x, y) u = F (x, y) F maps (x,y) to f'(x) (x, y) u = f '(x) the derivative of f(x) at x The slope of the tangent line at (x, f(x)) f '(x) = F (x, y) F maps (x,y) to f'(x) Direction Field Slope Field A 2-d plot representing f'(x) by a lineal element at (x, y) The slope of the tangent line at (x, f(x)) Separable Equations (1A) 13
Direction Field, First Order ODE u = F (x, y) u = F (x, y) F maps (x,y) to u (x, y) u = f '(x) y = f (x) f '(x) = F (x, y) F maps (x,y) to f'(x) the derivative of f(x) at x The slope of the tangent line at (x, f(x)) First Order ODE Find solution y=f(x) = g(x, y) where the first derivative y' is given by some formula g(x,y) containing variable x, y Direction Field Slope Field A 2-d plot of y'=f'(x) at (x, y) g(x, y) = Now, it also can be viewed as a function g maps (x,y) to f'(x) Separable Equations (1A) 14
Euler's Method F (x, y) = f '(x) y = f (x) h h h (x 0, y 0 ) (x 1, y 1 ) x 1 x 0 = dx = h y 1 y 0 = dy = f '(x)dx = F(x 0, y 0 )dx y 1 = y 0 + F(x 0, y 0 )h = f '(x)dx y i+1 = y i + F(x i, y i )h Separable Equations (1A) 15
Special Cases of Separable Equations = g 1 (x) = sin (x) d = g 2 ( y) y = sin ( y) the only literal y Autonomous ODEs attractors repellers attractors cos(x) + C repellers repellers attractors solution curves y(x): translated in the y direction solution curves y(x): translated in the x direction Separable Equations (1A) 16
Autonomous First Order ODEs = ( y 1)( y+1) h( y) h(y) y ' (x) = h( y) > 0 y ' (x) = h( y) < 0 y ' (x) = h( y) > 0 y > +1 1 < y < +1 y < 1 y y (x) y=+1 y ' (x) > 0 y ' (x) < 0 x increasing y (x) decreasing y( x) y= 1 y ' (x) > 0 increasing y (x) Separable Equations (1A) 17
Critical Points of Autonomous ODEs = ( y 1)( y+1) h( y) h(y) h( y) = 0 y = +1 ( y 1)( y+1) = 0 y = 1 y critical points (equilibrium, stationary points) +1, 1 constant solutions y = +1 y = 1 y=+1 y= 1 Separable Equations (1A) 18
Examples of Autonomous ODEs (1) = ( y 1)( y+1) ln y 1 y +1 = 2 x + c 2 y 1 y+1 = e2 x + c 2 = c 3 e 2 x 1 dx = 1dx ( y 2 1) 1 dy = 1 dx ( y 2 1) 1 1 2 ( ( y 1) 1 dy = 1 dx ( y +1)) assumed ( y 1) ( y +1) y 1 y+1 = +c 3 e 2 x y 1 y+1 = c 3 e 2 x y = 1+c 3 e2 x 1 c 3 e 2 x y = 1 c e 2x 3 1+c 3 e 2 x +c 3 c c 3 c 1 2 ( 1 ( y 1) dy 1 ( y +1) dy ) = 1 dx y = 1+c e2 x 1 c e 2 x 1 2 (ln y 1 ln y+1 ) = x + c 1 ln y 1 y +1 = 2 x + c 2 constant solutions y = +1 y = 1 c 0 : excluded in this method Separable Equations (1A) 19
Examples of Autonomous ODEs (2) y = 1+e2 x 1 e 2 x y = 1 e 2 x 1+e 2x Separable Equations (1A) 20
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