Math: Euler's Method for Differential Equations Worksheet

Question 1

Use Euler's Method with step size h = 0.5 to approximate y(1) for the differential equation dy/dx = x + y, given y(0) = 1.

Accepted Answer

Starting at (0, 1), the slope is 0 + 1 = 1, so y(0.5) is approximately 1 + 0.5(1) = 1.5. At (0.5, 1.5), the slope is 0.5 + 1.5 = 2, so y(1) is approximately 1.5 + 0.5(2) = 2.5.

Question 2

Use Euler's Method with step size h = 0.25 to approximate y(0.5) for dy/dx = 2x - y, given y(0) = 1.

Accepted Answer

At (0, 1), the slope is 2(0) - 1 = -1, so y(0.25) is approximately 1 + 0.25(-1) = 0.75. At (0.25, 0.75), the slope is 2(0.25) - 0.75 = -0.25, so y(0.5) is approximately 0.75 + 0.25(-0.25) = 0.6875.

Question 3

A table for dy/dx = y - x starts with x = 0 and y = 2. Complete the next two Euler steps using h = 1, and find the approximate value of y(2).

Accepted Answer

At (0, 2), the slope is 2 - 0 = 2, so y(1) is approximately 2 + 1(2) = 4. At (1, 4), the slope is 4 - 1 = 3, so y(2) is approximately 4 + 1(3) = 7.

Question 4

Use Euler's Method with h = 0.1 to approximate y(0.2) for dy/dx = x squared + y, given y(0) = 1.

Accepted Answer

At (0, 1), the slope is 0 squared + 1 = 1, so y(0.1) is approximately 1 + 0.1(1) = 1.1. At (0.1, 1.1), the slope is 0.1 squared + 1.1 = 1.11, so y(0.2) is approximately 1.1 + 0.1(1.11) = 1.211.

Question 5

For dy/dx = 3 - y with y(0) = 0, use Euler's Method with step size h = 0.5 to approximate y(1.5).

Accepted Answer

At (0, 0), the slope is 3, so y(0.5) is approximately 1.5. At (0.5, 1.5), the slope is 1.5, so y(1) is approximately 2.25. At (1, 2.25), the slope is 0.75, so y(1.5) is approximately 2.625.

Question 6

A student uses Euler's Method for dy/dx = x - 2y with y(0) = 1 and h = 0.5. The student writes y(0.5) = 1 + 0.5(0 - 2) = 0. Is the student's first step correct? Explain.

Accepted Answer

The student's first step is correct. At the starting point (0, 1), the slope is 0 - 2(1) = -2, so y(0.5) is approximately 1 + 0.5(-2) = 0.

Question 7

Use the slope field for dy/dx = x + y and the starting point (0, 1). With h = 0.5, draw the first two Euler line segments and state the approximate value of y(1).

Accepted Answer

The first slope at (0, 1) is 1, so the first segment goes to approximately (0.5, 1.5). The second slope at (0.5, 1.5) is 2, so the next segment goes to approximately (1, 2.5). Therefore, y(1) is approximately 2.5.

Question 8

Use Euler's Method with h = 0.2 to approximate y(0.4) for dy/dx = y cos(x), given y(0) = 2. Round your final answer to three decimal places.

Accepted Answer

At (0, 2), the slope is 2 cos(0) = 2, so y(0.2) is approximately 2 + 0.2(2) = 2.4. At (0.2, 2.4), the slope is 2.4 cos(0.2), which is approximately 2.352. Therefore, y(0.4) is approximately 2.4 + 0.2(2.352) = 2.870.

Question 9

The exact solution of a differential equation is y = e^x, with y(0) = 1. Euler's Method is used on dy/dx = y with h = 0.5 to estimate y(1). Find the Euler estimate and compare it to the exact value e, using e approximately 2.718.

Accepted Answer

At (0, 1), the slope is 1, so y(0.5) is approximately 1.5. At (0.5, 1.5), the slope is 1.5, so y(1) is approximately 2.25. The exact value is about 2.718, so the Euler estimate is 0.468 too low.

Question 10

Explain why using a smaller step size often gives a better Euler's Method approximation. Use the idea of tangent line segments in your answer.

Accepted Answer

A smaller step size often gives a better approximation because each tangent line segment is used for a shorter interval. This usually reduces the amount of error caused by following a straight line while the true solution curve is bending.

Math: Euler's Method for Differential Equations

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