Question

Use Backward Euler’s Method to find the approximate values of the solution of the following differential equation at t=0.12, 0.23, 0.37, 0.45, and 0.72. Use a step size of h=0.08.

y′+2y=2−e⁻⁴ᵗ y(0)=1

Answer 1

The student's question on using Backward Euler's Method for solving a differential equation cannot be addressed with the provided materials, which do not include required formulas or context for the specified equation and initial condition.

Step-by-step explanation:

The question involves using the Backward Euler's Method to approximate values of a solution to a differential equation at specified time points with a given step size. This method is a numerical technique used to solve ordinary differential equations (ODEs) and is particularly useful for stiff equations. Unfortunately, the materials provided do not contain pertinent information or equations to solve the student's specific question regarding the differential equation y' + 2y = 2−e⁻´ᵗ with the initial condition y(0) = 1.

With the Backward Euler's Method, one typically rearranges the equation to express y at the next time step in terms of y at the current step, iteratively. This process includes guessing an initial value for y at the next time step, substituting it back into the rearranged equation, solving for a new y, and repeating until y changes insignificantly.

Since the materials provided mention the improvement of y from 0.012 to 0.011, this suggests that an iterative refinement process was utilized, but without the correct context or equations, it is difficult to apply this information directly to the student's question.