Consider the differential equation

dydx4x2yfracdydx 4x2 y

for the initial condition y3y 3 at x1x 1 the value of yy at x14x 14 obtained using eulers method with a stepsize of
02 is round off to one decimal place

Question

Consider the differential equation:

$\frac{d y}{d x} = 4 (x + 2) - y$

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 ______. (round off to one decimal place)

anh · Answer

Given Data:
- Differential equation: $\frac{d y}{d x} = 4 (x + 2) - y$
- Initial condition: $y (1) = 3$
- Step-size $h = 0.2$
- We want $y$ at $x = 1.4$
Euler's Method: Euler's method formula:
$y_{n + 1} = y_{n} + h \cdot f (x_{n}, y_{n})$
Step 1: From $x = 1$ to $x = 1.2$ :
$f (1, 3) = 4 (1 + 2) - 3 = 12 - 3 = 9$ $y_{1.2} = 3 + 0.2 \cdot 9 = 3 + 1.8 = 4.8$
Step 2: From $x = 1.2$ to $x = 1.4$ :
$f (1.2, 4.8) = 4 (1.2 + 2) - 4.8 = 4 \times 3.2 - 4.8 = 12.8 - 4.8 = 8$ $y_{1.4} = 4.8 + 0.2 \cdot 8 = 4.8 + 1.6 = 6.4$

The value of $y$ at $x = 1.4$ is approximately 6.4.

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