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Use a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05.
y' = y - y^2 , y(0) = 0.6; y(0.5)

Use a numerical solver and Eulers method to obtain a fourdecimal approximation of the indicated value First use h 01 and then use h 005 y y y2 y0 06 y05 class=

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sqdancefan

Answer:

  • y(0.5) ≈ 0.7137
  • y(0.5) ≈ 0.7129

Step-by-step explanation:

You want the value of y(0.5) using Euler's method with h = 0.1 and h = 0.05 and the differential equation y' = y -y², y(0) = 0.6.

Euler's method

Using Euler's method, the next value of y is ...

  y(x +h) ≈ y(x) +h(y(x) -y(x)²)

To find y(0.5), we need to use ...

  0.5/0.1 = 5 steps with h = 0.1, or

  0.5/0.05 = 10 steps with h = 0.05.

The attachment shows a solver repeatedly applying the above formula, starting with y = 0.6.

  • after 5 steps with h = 0.1, y(0.5) ≈ 0.7137
  • after 10 steps with h = 0.05, y(0.5) ≈ 0.7129

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Additional comment

The solution to the differential equation is ...

  [tex]y(x)=\dfrac{1}{1+\dfrac{2}{3}e^{-x}}\\\\y(0.5)\approx 0.712071[/tex]

500 steps with h = 0.001 gives a value of y(0.5) accurate to 4 decimal places (0.712087).

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