Introduction Applications of Differential Equations Checking Solutions and IVP

Introduction applications of differential equations

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Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Introduction to Maple 1 Introduction to Maple : A Symbolic Math Program We enter a function y ( t ) = 3 e - t cos(2 t ), y := t 3 · exp( - t ) · cos(2 · t ); The arrow is - and > and multiplication is * . To plot this function plot ( y ( t ) , t = 0 .. 2 · Pi); Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (43/47)
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The Class — Overview The Class... Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Introduction to Maple 2 We have the function: y ( t ) = 3 e - t cos(2 t ), This can be differentiated (and stored in variable dy ) by typing dy := diff ( y ( t ) , t ); Maple gives: dy := - 3 e - t cos(2 t ) - 6 e - t sin(2 t ) The absolute minimum and a relative maximum are found with Maple: tmin := fsolve ( dy = 0 , t = 1 .. 2); y ( tmin ); tmax := fsolve ( dy = 0 , t = 2 . 5 .. 3 . 5); y ( tmax ); The result was an absolute minimum at (1 . 33897 , - 0 . 703328). The result was a relative maximum at (2 . 90977 , 0 . 1462075). Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (44/47)
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The Class — Overview The Class... Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Introduction to Maple 3 With y ( t ) = 3 e - t cos(2 t ), we can solve Z 3 e - t cos(2 t ) dt and Z 5 0 3 e - t cos(2 t ) dt These can be integrated by typing int ( y ( t ) , t ); int ( y ( t ) , t = 0 .. 5); evalf (%); For the indefinite integral, Maple gives: - 3 5 e - t cos(2 t ) + 6 5 e - t sin(2 t ) For the definite integral, Maple gives: 3 5 - 3 5 e - 5 cos(10) + 6 5 e - 5 sin(10) = 0 . 59899347 Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (45/47)
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The Class — Overview The Class... Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Introduction to Maple 4 Show y ( t ) = 3 e - t cos(2 t ) is a solution of the differential equation y 00 + 2 y 0 + 5 y = 0 . The function and derivatives are entered by y := t 3 · exp( - t ) · cos(2 · t ); dy := diff ( y ( t ) , t ); sdy := diff ( y ( t ) , t $2); If we type sdy + 2 · dy + 5 · y ( t ); Maple gives 0 , which verifies this is a solution. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (46/47)
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The Class — Overview The Class... Introduction Applications of Differential Equations Checking Solutions and IVP Evaporation Example Nonautonomous Example Introduction to Maple Introduction to Maple 5 Maple finds the general solution to the differential equation de := diff ( Y ( t ) , t $2) + 2 · diff ( Y ( t ) , t ) + 5 · Y ( t ) = 0; dsolve ( de, Y ( t )); Maple produces Y ( t ) = C 1 e - t sin(2 t ) + C 2 e - t cos(2 t ) To solve an initial value problem, say Y (0) = 2 and Y 0 (0) = - 1, enter dsolve ( { de, Y (0) = 2 , D ( Y )(0) = - 1 } , Y ( t )); Maple produces Y ( t ) = 1 2 e - t sin(2 t ) + 2 e - t cos(2 t ) , which is made into a useable function by typing Y := unapply ( rhs (%) , t ); Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (47/47)
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