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Chapter 8: Problem 3

Consider the initial value problem \(y^{\prime}(t)=t^{2}-3 y^{2}, y(3)=1\) What is the approximation to \(y(3.1)\) given by Euler's method with a time step of \(\Delta t=0.1 ?\)

Short Answer

Expert verified
Answer: The approximate value of y(3.1) using Euler's method with a time step of 0.1 is 1.6.

Step by step solution

01

Understand Euler's method formula

Euler's method is used for approximating the solution of an initial value problem of the form \(y^{\prime}(t) = f(t, y(t))\) and an initial condition \(y(t_0) = y_0\), where t is the independent variable, and y is the dependent variable. The method advances the solution from one step to the next as follows: \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\) In our problem, we have the initial condition \(y(3)=1\), the differential equation \(y^{\prime}(t)=t^{2}-3y^{2}\), and the time step \(\Delta t=0.1\).
02

Set up the first iteration

In the first iteration of the Euler's method, we need to find the value of \(y(3 + \Delta t)=y(3.1)\) using the initial condition \(y(3)=1\) and the given differential equation \(y^{\prime}(t)=t^{2}-3y^{2}\). We will use the formula: \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\) where \(y_n=y(3)=1\), \(t_n=3\), \(f(t_n, y_n)=f(3,1)=3^{2}-3(1)^{2}=9-3\), and \(\Delta t=0.1\).
03

Calculate y(3.1) using Euler's method

Using the formula and the values from Step 2, we can now calculate the approximate value of y(3.1): \(y_{n+1} = y_n + \Delta t \cdot f(t_n, y_n)\) \(y(3.1) = y(3) + \Delta t \cdot f(3,1)\) \(y(3.1) = 1 + 0.1 \cdot (9-3)\) \(y(3.1) = 1 + 0.1 \cdot 6\) Now, we can calculate the value of y(3.1): \(y(3.1) = 1 + 0.6\) \(y(3.1) = 1.6\) The approximation of y(3.1) using Euler's method with a time step of \(\Delta t=0.1\) is 1.6.

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