Exercises for Question Type 2: Applying Euler's method to second order differential equations

Question 1

Apply Euler's method with h = 0.5 to approximate x (t) and \overset{x}{˙} (t) at t = 2.5 for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 . Present your results in a table with columns t_{n}, x_{n}, \overset{x}{˙}_{n} .

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Question 2

Use Euler's method with step length h = 0.5 to approximate x (2.5) and \overset{x}{˙} (2.5) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 .

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Question 3

This question assesses the application of Euler's method to approximate solutions for a second-order differential equation by transforming it into a system of two first-order differential equations.

Use Euler's method with step length h = 0.3 to approximate x (1.5) and \overset{x}{˙} (1.5) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 .

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Question 4

Use Euler's method with step length h = 0.2 to approximate x (1.0) and \overset{x}{˙} (1.0) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 . Then comment on the qualitative behavior of the approximate solution as t increases.

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Question 5

Derive the general Euler iteration formulas for a second order ODE \frac{d ^{2} x}{d t ^{2}} = f (t, x, \overset{x}{˙}) by converting it into a system of first order equations. Then specify these formulas for f (t, x, y) = 2 x - y + t^{2} .

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Question 6

Use Euler's method with step length h = 0.1 to approximate x (t) and \overset{x}{˙} (t) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4, \overset{x}{˙} (0) = 2 . Show only the first three iterations (up to t = 0.3) and state your approximate values at t = 0.3 .

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Question 7

Use Euler's method with step length h = 0.4 to approximate x (2.0) and \overset{x}{˙} (2.0) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 .

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Question 8

Use Euler's method with step length h = 0.1 to approximate x (0.5) and \overset{x}{˙} (0.5) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 .

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Question 9

Compare the approximations of x (0.5) and \overset{x}{˙} (0.5) obtained by Euler's method with step sizes h = 0.1 and h = 0.25 for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 . Discuss how the step size affects the numerical accuracy.

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Question 10

Mathematic HL/Further Mathematics

Use Euler's method with step length h = 0.2 to approximate x (1.0) and \overset{x}{˙} (1.0) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 .

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Question 11

Use Euler's method with step length h = 0.25 to approximate x (1.25) and \overset{x}{˙} (1.25) for the differential equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4 and \overset{x}{˙} (0) = 2 .

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Question 12

Outline an algorithm to approximate x (2.5) and \overset{x}{˙} (2.5) using Euler's method with h = 0.1 for the equation \frac{d ^{2} x}{d t ^{2}} = 2 x - \frac{d x}{d t} + t^{2} given x (0) = 4, \overset{x}{˙} (0) = 2 . State how many steps are required and how you would implement the iteration in a spreadsheet or program.

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IB Mathematics Applications & Interpretation Question Type 2: Applying Euler's Method to Second Order Differential Equations Exercises