Use Euler's method with step size $0.25$ to estimate $y ( 1 )$, where $y ( x )$ is the solution of the initial-value problem. Round your answer to four decimal places.
$$y ^ { t } = 4 x + y ^ { 2 } , y ( 0 ) = 0$$
A) $y ( 1 ) = 4.1622$
B) $y ( 1 ) = 1.6622$
C) $y ( 1 ) = 3.1622$
D) $y ( 1 ) = 3.1622$.
E) $y ( 1 ) = 2.6622$.

Question

Quizplus · Accepted Answer

Euler's method involves using the formula $y_{n+1} = y_n + hf(x_n, y_n)$ to iteratively calculate the value of $y$ at successive points, starting from an initial condition. Here, $h = 0.25$, $f(x, y) = 4x + y^2$, and the initial condition is $y(0) = 0$. Calculating step by step for $x = 0.25, 0.5, 0.75, 1$, we find $y(1) \approx 1.6622$.

Use Euler's Method with Step Size 0.250.250.25 To Estimate y(1)y ( 1 )y(1)

Use Euler's Method with Step Size $0.25$ To Estimate $y ( 1 )$