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 Damped Spring-Mass Problem 1 Damped Spring-Mass Problem: Assume a mass attached to a spring with resistance satisfies the second order linear differential equation y 00 ( t ) + 2 y 0 ( t ) + 5 y ( t ) = 0 Skip Example Show that one solution to this differential equation is y 1 ( t ) = 2 e - t sin(2 t ) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (32/47)
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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 Damped Spring-Mass Problem 2 Solution: Damped spring-mass problem The 1 st derivative of y 1 ( t ) = 2 e - t sin(2 t ) y 0 1 ( t ) = 2 e - t (2 cos(2 t )) - 2 e - t sin(2 t ) = 2 e - t (2 cos(2 t ) - sin(2 t )) The 2 nd derivative of y 1 ( t ) = 2 e - t sin(2 t ) y 00 1 ( t ) = 2 e - t ( - 4 sin(2 t ) - 2 cos(2 t )) - 2 e - t (2 cos(2 t ) - sin(2 t )) = - 2 e - t (4 cos(2 t ) + 3 sin(2 t )) Substitute into the spring-mass problem y 00 1 + 2 y 0 1 + 5 y = - 2 e - t (4 cos(2 t ) + 3 sin(2 t )) +2(2 e - t (2 cos(2 t ) - sin(2 t ))) + 5(2 e - t sin(2 t )) = 0 It is often easy to check that a solution satisfies a differential equation. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (33/47)
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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 Damped Spring-Mass Problem 3 Graph of Damped Oscillator 0 π/2 π 3π/2 -0.2 0 0.2 0.4 0.6 0.8 1 t y(t) Damped Spring - y(t) = 2 e -t sin(2t) Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (34/47)
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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 Initial Value Problem Definition (Initial Value Problem) An initial value problem for an n th order differential equation y ( n ) = f ( t, y, y 0 , y 00 , ..., y ( n - 1) ) on an interval I consists of this differential equation together with n initial conditions y ( t 0 ) = y 0 , y 0 ( t 0 ) = y 1 , ..., y ( n - 1) ( t 0 ) = y n - 1 prescribed at a point t 0 I , where y 0 , y 1 , ..., y n - 1 are given constants. Under reasonable conditions the solution of an Initial Value Problem has a unique solution. Joseph M. Mahaffy, h [email protected] i Lecture Notes – Introduction to Differential Eq — (35/47)
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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 Evaporation Example 1 Evaporation Example: Animals lose moisture proportional to their surface area Skip Example If V ( t ) is the volume of water in the animal, then the moisture loss satisfies the differential equation dV dt = - 0 . 03 V 2 / 3 , V (0) = 8 cm 3 The initial amount of water is 8 cm 3 with t in days Verify the solution is V ( t ) = (2 - 0 . 01 t ) 3 Determine when the animal becomes totally desiccated according to this model Graph the solution Joseph M. Mahaffy, h [email protected]
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