Exercise 5.2 [Euler and Modified Euler]

EXERCISE 5.2 [Euler and Modified Euler]

1. Use Euler's method to find y (0.4) given y' = xy, y (0) = 1.

[Ans. 1.0611]

2. Use Modified Euler's method and obtain y (0.2) given

dy/dx = log (x + y), y (0) = 1, h = 0.2

[Ans. y (0.2) = 1.0095]

3. Using modified Euler's method, get y (0.2), y(0.4), y (0.6) given

dy/dy = y - x², y (0) = 1.

 [Ans. 1.218, 1.467, 1.737]

4. Using modified Euler's method, solve dy/dx  = y + x², y(0) = 1 to find y (0.2) and y (0.4) correct to 3 decimals.

[Ans. y (0.2) = 1.24, y (0.4) = 1.514]

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5. Given dy/dx – x² – y = 4, y(0) = 1. Determine y (0.02), y (0.04) and y (0.06) using Euler's modified method.

 [Ans. y (0.02) 1.0202, y (0.04) = 1.0408, y (0.06) = 1.0619]

6. Solve, dy/dx = 2y / x + x³ to obtain y (1.2) and y (1.4) given y = 0.5, when x = 1 by modified Euler's method.

 [Ans. y (1.2) = 1.0228, y (1.4) = 1.8847]

7. Using modified Euler's method solve the differential equation

dy/dx + y/x = y² given that y = 1 when x = 1. Choose the step length 0.1. Find y at x = 16.

[Ans. y (1.6) = 1.1766]

8. Given = dy/dx x + y² and y (0) = 1. Find an approximate value of y at x = 0.5 by modified Euler's method. Taking h = 0.1

 [Ans. y (0.5) = 2.2070]

9. Given dy/dx = x – y / x + y y(2) =1 Find y (1) by taking h = -0.2  using Euler's modified method.

 [Ans. y (1) = 0.73207]

10. Solve, dy/dx = x √1 + y2, y(1) = 0 at x = 3 taking h = 0.4 using Euler's modified method.

[Ans. y (3) = 21.671]

11. Given dy/dx = y/x – (5/2) x² y³, y(1) = 1 / √2 .Find y (2) with h = 0.125 using Euler's modified method.

[Ans. y (1) = 0.34939]

12. Solve the differential equation dy / dx = -xy², y (0) = 2 by modified

Euler's method and obtain y (0.2) by taking h = 0.1

[Ans. y (0.2) = 1.9227]

13. Solve dy/dx = 1 - y with y (0) = 0 at x = 0.1, 0.2, 0.3 using modified Euler's method.

 [Ans. y (0.1) = 0.095, y (0.2) = 0.1890, y (0.3) = 0.2587]