Exercises for Question Type 4: Approximating values for coupled system using Euler's method with fixed iteration and step length using technology

Question 1

Use Euler's method with step size h = 0.1 to approximate the solution at t = 0.5 for the system of differential equations: \frac{d x}{d t} = x^{2} + y^{2} \frac{d y}{d t} = 2 x - y + t^{2} given the initial conditions x (0) = 4 and y (0) = 2 .

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Question 2

Using Euler’s method with a step size of h = 0.1, approximate the values of x (0.3) and y (0.3) for the system of coupled differential equations: \frac{d x}{d t} \frac{d y}{d t} = x^{2} + y^{2} = 2 x - y + t^{2} with the initial conditions x (0) = 4 and y (0) = 2 .

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Question 3

The question requires the application of Euler's method for a system of coupled first-order differential equations over two steps to estimate values at t = 0.5 .

Use Euler’s method with step length h = 0.25 to find x (0.5) and y (0.5) for the system of differential equations: \frac{d x}{d t} = x^{2} + y^{2} \frac{d y}{d t} = 2 x - y + t^{2} given the initial conditions x (0) = 4 and y (0) = 2 .

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Question 4

Approximate x (1.0) and y (1.0) using Euler’s method with step size h = 0.5 for the system of differential equations \frac{d x}{d t} = x^{2} + y^{2} and \frac{d y}{d t} = 2 x - y + t^{2}, given the initial conditions x (0) = 4 and y (0) = 2 .

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Question 5

Euler's Method for Systems of Differential Equations

Use Euler's method with two steps of size h = 1.0 to approximate x (2.0) and y (2.0) for the system of differential equations \frac{d x}{d t} = x^{2} + y^{2}, \frac{d y}{d t} = 2 x - y + t^{2} with initial conditions x (0) = 4 and y (0) = 2 .

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Question 6

Estimate x (0.75) and y (0.75) by Euler’s method with step size h = 0.25 for the system of differential equations \frac{d x}{d t} = x^{2} + y^{2} \frac{d y}{d t} = 2 x - y + t^{2} given the initial conditions x (0) = 4 and y (0) = 2 .

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Question 7

With step size h = 0.5, apply Euler’s method to approximate x (1.5) and y (1.5) for the system of differential equations: \frac{d x}{d t} = x^{2} + y^{2}, \frac{d y}{d t} = 2 x - y + t^{2} given the initial conditions x (0) = 4 and y (0) = 2 .

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Question 8

Starting from x (0) = 4, y (0) = 2, use Euler’s method with h = 0.2 to approximate x (0.4) and y (0.4) for \frac{d x}{d t} = x^{2} + y^{2} and \frac{d y}{d t} = 2 x - y + t^{2} .

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Question 9

Using a step size of h = 0.25, apply Euler’s method to approximate x (1.0) and y (1.0) for the system of differential equations \frac{d x}{d t} = x^{2} + y^{2} \frac{d y}{d t} = 2 x - y + t^{2} with the initial conditions x (0) = 4 and y (0) = 2 .

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Question 10

Numerical solutions to differential equations using Euler's method for a system of first-order ordinary differential equations.

Apply Euler’s method with step size h = 0.1 to estimate x (0.4) and y (0.4) for the system \frac{d x}{d t} = x^{2} + y^{2}, \frac{d y}{d t} = 2 x - y + t^{2}, starting at (0, 4, 2) . Give your final answers to three significant figures.

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Question 11

For the system of differential equations \frac{d x}{d t} = x^{2} + y^{2} \frac{d y}{d t} = 2 x - y + t^{2} with initial conditions x (0) = 4 and y (0) = 2, use Euler’s method with step length h = 0.1 to approximate the values of x (0.2) and y (0.2) .

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IB Mathematics Applications & Interpretation Question Type 4: Approximating Values for Coupled System Using Euler's Method with Fixed Iteration and Step Length Using Technology Exercises