Use Euler's method to calculate the first three...

Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Calculate the exact solution and investigate the accuracy of your approximations. Round your results to four decimal places. $y^{\prime}=y e^{x}, \quad y(0)=2, \quad d x=0.5$

Key Concepts

Euler's Method

Euler's Method is a numerical technique used to approximate solutions to ordinary differential equations (ODEs) with a given initial value. It operates by stepping forward a small increment in the independent variable and using the derivative information to estimate the new value of the function, thereby constructing an approximate solution along the domain.

Initial Value Problem

An Initial Value Problem (IVP) for ODEs involves finding a function that satisfies the differential equation and also meets a specified value at a given point. This additional piece of information, usually given as y(x0) = y0, allows for the determination of a unique solution to the differential equation.

Exact Solution

The exact solution of an ODE refers to a closed-form expression or function that precisely satisfies both the differential equation and the given initial condition. It serves as a benchmark to assess the accuracy of numerical methods such as Euler's Method.

Step Size

The step size, often denoted as ?x or h, represents the incremental change in the independent variable used in numerical methods. It directly influences the approximation's accuracy, with smaller step sizes generally leading to more accurate results but requiring more computational effort.

Error Analysis

Error analysis in the context of numerical solutions involves comparing the approximated results from methods like Euler's Method to the exact solution of the ODE. This analysis helps in understanding the limitations and convergence properties of the numerical method, including truncation errors and propagation of error over successive steps.

Transcript

00:01 For this question, we're given that y prime is equal to y times e to the x, y of 0 is equal to 2, and dx is equal to 0 .5.

00:08 And it does have a mark on here to use your calculator.

00:11 So this question is basically asking us to do two things.

00:14 First, we're supposed to use euler's method to calculate the first three approximations.

00:19 And then after that, we're going to calculate the exact solution and kind of compare for accuracy.

00:26 So first things, first, we need to remember what exact.

00:30 The oler's method is.

00:32 So oro's method that's just telling us that y to the n is just going to equal to yn minus 1 plus the function xn minus 1, yn minus 1 times dx.

00:52 And so then we also need to know how to find our xns, and our xns is just going to be xn minus 1 plus dx.

01:03 And so this is is an initial value problem because we are given our initial value right here where this zero is our x not and this two is r so we're going to implement this to find our y1 y2 and y3 for this problem so my first step is i am going to find my x1 and x2 first so i already know that x not is just equal to zero my x1 is going to equal to zero plus d x1 is going to equal to zero plus which is 0 .5 and then x2 is going to equal to x1 plus dx 0 .5 so x2 is just going to equal to 1.

01:54 And now next i am going to find my y1 2 and 3 so y1 following this formula is just going to equal to y not plus my function at x not y not so that looking at this function here that's going to be y0 e to the x not and then times dx.

02:19 So plugging my values, i already know that y not is just equal to 2.

02:23 So i get 2 plus 2 times e to the x not, which we know is 0 times the x which we're given is 0 .5...

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