If \(\frac{dy}{dx}\) = x - y² and y(0) = 1, then y(0.1) correct upto two decimal places (approx.) is:

Concept:

Euler's Method to generate a numerical solution to an initial value problem of the form:

y' = f (x, y), y(x₀) = y₀

y_{n + 1} = y_n + h f(x_n, y_n)

Calculation:

We have,

x₀ =  0, y₀ = 1 , h = 0.1

x₁ = x₀ + h

For n = 0

⇒ y₁ = y₀ + hf(x_n, y_n)

⇒ y₁ = 1 - 0.1 ×  (0 - 1)²

⇒ y₁ = 1 - 0.1

⇒ y₁ = 0. 9

For n = 1

⇒ y₂ = y₁ + hf(x₁ - y₁²)

⇒ y₂ = 0.9 + 0.1[0.1 - (0.9)²]

⇒ y₂ = 0.9 + 0.1[0.1 - 0. 81]

⇒ y₂ = 0.9 - 0.1 ×  0.71

⇒ y₂ = 0.9 - 0.071

⇒ y₂ = 0. 829

⇒ y₂ = 0.83

∴ The approximately. two decimal place value is 0.83