Please solve & show steps... ### Consider the initial value problem\[ y' = \frac{y^2 + 2ty}{3 + t^2}, \quad y(1) = 2. \]Use Euler's method with \( h = 0.1, 0.05, 0.025, \) and \( 0.01 \) to explore the solution of this problem for \( 1 \leq t \leq 3 \).**What is your best estimate at \( t = 3 \)?**- ○ 112.84- ○ 38.67- ○ 238.44- ○ 65.34- ○ No reliable estimate is possible.#### Explanation:The problem involves solving a first-order ordinary differential equation (ODE) using Euler's method, which is an iterative numerical technique. Euler's method uses a step size \( h \) to approximate the solution of the ODE over a given interval.For each step of Euler's method:\[ y_{n+1} = y_n + h f(t_n, y_n) \]where \( f(t, y) \) is the right-hand side of the ODE, \( h \) is the step size, \( t_n \) is the current time/position, and \( y_n \) is the current solution value.In this problem, you are to apply Euler's method using different step sizes (\( h = 0.1, 0.05, 0.025, 0.01 \)) to approximate the solution from \( t = 1 \) to \( t = 3 \). Then, you need to determine the best estimate for \( y \) at \( t = 3 \) based on these approximations and choose from the given options.

Name: Please solve & show steps... ### Consider the initial value problem\[
Uploaded: 2024-03-20T06:38:44.000Z
Channel: Bartleby
Description: Please solve & show steps... ### Consider the initial value problem\[ y' = \frac{y^2 + 2ty}{3 + t^2}, \quad y(1) = 2. \]Use Euler's method with \( h = 0.1, 0.05, 0.025, \) and \( 0.01 \) to explore the solution of this problem for \( 1 \leq t \leq 3