Question

Consider the differential equation \[ \frac{dy}{dx} = 4(x + 2) - y \] For the initial condition \( y = 3 \) at \( x = 1 \), the value of \( y \) at \( x = 1.4 \) obtained using Euler's method with a step-size of 0.2 is \(\underline{\hspace{2cm}}\) . (round off to one decimal place)

Hint:

Euler's method approximates the solution of a differential equation by using a simple iterative formula based on the function's derivative at each point.

Answer 1

Correct Answer: 6.3

Euler's method for solving a differential equation is given by:
\[ y_{n+1} = y_n + h \cdot f(x_n, y_n) \] where \( h \) is the step size, \( f(x_n, y_n) \) is the derivative at point \( (x_n, y_n) \), and \( y_{n+1} \) is the next value of \( y \).
Given \( \frac{dy}{dx} = 4(x + 2) - y \), we can write the function as:
\[ f(x, y) = 4(x + 2) - y. \] We start with the initial condition \( y_1 = 3 \) at \( x = 1 \), and the step size \( h = 0.2 \). Now, we apply Euler's method iteratively:
For \( x_1 = 1 \) and \( y_1 = 3 \):
\[ f(1, 3) = 4(1 + 2) - 3 = 12 - 3 = 9. \] Thus, \( y_2 = y_1 + h \cdot f(1, 3) = 3 + 0.2 \times 9 = 3 + 1.8 = 4.8 \). For \( x_2 = 1.2 \) and \( y_2 = 4.8 \):
\[ f(1.2, 4.8) = 4(1.2 + 2) - 4.8 = 4(3.2) - 4.8 = 12.8 - 4.8 = 8. \] Thus, \( y_3 = y_2 + h \cdot f(1.2, 4.8) = 4.8 + 0.2 \times 8 = 4.8 + 1.6 = 6.4 \). For \( x_3 = 1.4 \) and \( y_3 = 6.4 \):
\[ f(1.4, 6.4) = 4(1.4 + 2) - 6.4 = 4(3.4) - 6.4 = 13.6 - 6.4 = 7.2. \] Thus, \( y_4 = y_3 + h \cdot f(1.4, 6.4) = 6.4 + 0.2 \times 7.2 = 6.4 + 1.44 = 7.84 \). Therefore, the value of \( y \) at \( x = 1.4 \) using Euler's method is \( \boxed{7.8} \).