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Unformatted text preview: MAT 2384Practice Problems on Numerical Methods for Diﬀerential Equations Question 1. For each of the Following IVP’s, apply Euler Method with the given step size h to estimate solutions on the given interval. Round your answers to 6 decimal places. Then Solve the IVP exactly and compare your estimations with the Exact values. 1. y + 5x4 y 2 = 0, y (0) = 1, h = 0.1 on [0, 0.5]
1 2. y = 2 π 1 − y 2 , y (0) = 0, h = 0.1 on [0, 0.3] 3. y = (y + x)2 , y (0) = 0, h = 0.1 on [0, 0.4] Question 2. For each of the Following IVP’s, apply the Improved Euler Method with the given step size h to estimate solutions on the given interval. Round your answers to 6 decimal places. Then Solve the IVP exactly and compare your estimations with the Exact values. 1. y = y − y 2 = 0, y (0) = 0.5, h = 0.1 on [0, 0.3] 2. y + 2xy 2 = 0, y (0) = 1, h = 0.2 on [0, 0.6] 3. y = 2(y 2 + 1), y (0) = 0, h = 0.05 on [0, 0.2] Question 3. Consider the IVP: y = 2x−1 y − ln x + x−1 , y (1) = 0. (1) Verify that the Exact solution is y = (ln x)2 + ln x (2) Use the Improved Euler Method to estimate solutions of the IVP on the interval 1 ≤ x ≤ 1.6 using a step size of h = 0.2. Round your answers to 6 decimal places (3) Use the RungeKutta method of order 4 to estimate solutions of the IVP on the interval 1 ≤ x ≤ 1.6 using a step size of h = 0.2. Round your answers to 6 decimal places 1 (4) Make a table to compare your estimates in parts (2) and (3) with the exact values (from part (1)). Question 4. Use the RungeKutta method of order 4 to estimate solutions of the IVP y = xy + cos x, y (0) = 0 on the interval 0 ≤ x ≤ 0.6 using a step size of h = 0.2. 2 ...
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 Spring '10
 khoury
 Numerical Analysis, Equations, IVP, step size, Runge–Kutta methods

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