IVP-Exercises-Solutions

# IVP-Exercises-Solutions - dy dt = p 2 y-y 2 y = 1 Use...

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Reduce the third-order initial-value problem y 000 = y 00 - 3 ty 0 + y + 1 , y ( 0 ) = 0 , y 0 ( 0 ) - y 00 ( 0 ) = a , y 0 ( 0 ) + y 00 ( 0 ) = b to a system of ﬁrst-order differential equations (assume a , b constants). Reduce the system of second-order equations u 00 = u 0 + v + 1 , v 00 = u - v 0 , u ( 0 ) = 1 , u 0 ( 0 ) = 0 , v ( 0 ) = 0 , v 0 ( 0 ) = 0 to a system of ﬁrst-order differential equations.

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Use Euler’s method with step-size h = 0 . 1 to approximate y ( 0 . 2 ) where y is the solution of the initial-value problem dy dt = p 2 y - y 2 , y ( 0 ) = 1 . Use the midpoint method with step-size h = 0 . 1 to approximate y ( 0 . 2 ) where y is the solution of the initial-value problem dy dt = p 2 y - y 2 , y ( 0 ) = 1 .
Use the modiﬁed Euler method with step-size h = 0 . 1 to approximate y ( 0 . 2 ) where y is the solution of the initial-value problem dy dt = p 2 y - y 2 , y ( 0 ) = 1 . Use Heun’s method with step-size h = 0 . 1 to approximate y ( 0 . 2 ) where y is the solution of the initial-value problem dy dt = p 2 y - y 2 , y ( 0 ) = 1 .

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Use the classical Runge-Kutta method of order 4 with step-size h = 0 . 1 to approximate y ( 0 . 2 ) where y is the solution of the initial-value problem
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Unformatted text preview: dy dt = p 2 y-y 2 , y ( ) = 1 . Use Heun’s method with step-size h = . 5 to approximate y ( 1 . ) where y is the solution of the initial-value problem dy dt =-y ln y , y ( ) = 1 2 . Consider the IVP system y = ± u v ² = ± t + u + v 1 1 + u + v ² , y ( ) = ± u ( ) v ( ) ² = ± 1 1 ² . Use Euler’s method with time-step h = 1 / 4 to compute y 2 ' y ( t 2 ) = y ( 1 / 2 ) . Consider the IVP system y = ± u v ² = ± u + 2 v 3 u + 2 v ² , y ( ) = ± u ( ) v ( ) ² = ± 6 4 ² . Use the classical Runge-Kutta method of order 4 with time-step h = . 1 to compute y 2 ' y ( t 2 ) = y ( . 2 ) ....
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IVP-Exercises-Solutions - dy dt = p 2 y-y 2 y = 1 Use...

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