Suppose $\frac { d y } { d x } = y - x ^ { 2 }$ , $y ( 0 ) = 3$ . Use Euler's method with step size $h = 0.2$ to approximate $y ( 0.4 )$ .
A)3.6
B)3.2
C)3.4
D)5.14
E)1
F)4.312
G)4
H)6.1

Question

Quizplus · Accepted Answer

Euler's method formula is $y_{n+1} = y_n + h f(x_n, y_n)$, where $f(x, y) = \frac{dy}{dx}$. Starting with $x_0 = 0$, $y_0 = 3$, and $h = 0.2$:1. Calculate $y_1$:   - $f(x_0, y_0) = 3 - 0^2 = 3$   - $y_1 = y_0 + h \cdot f(x_0, y_0) = 3 + 0.2 \cdot 3 = 3.6$2. Calculate $y_2$ at $x_2 = 0.4$:   - $f(x_1, y_1) = 3.6 - 0.2^2 = 3.6 - 0.04 = 3.56$   - $y_2 = y_1 + h \cdot f(x_1, y_1) = 3.6 + 0.2 \cdot 3.56 = 3.6 + 0.712 = 4.312$Therefore, using Euler's method, $y(0.4) \approx 4.312$.

Suppose dydx=y−x2\frac { d y } { d x } = y - x ^ { 2 }dxdy​=y−x2

Suppose $\frac { d y } { d x } = y - x ^ { 2 }$