Question

Estimate the following solutions using Euler’s method with n=5 steps over the interval t=[0,1]. If you are able to solve the initial-value problem exactly, compare your solution with the exact solution. If you are unable to solve the initial-value problem, the exact solution will be provided for you to compare with Euler’s method. How accurate is Euler’s method? y′=y+t², y(0)=3. Exact solution is y=5eᵗ−2−t²−2t (Use euler's method only)

Answer 1

Euler's method is a numerical method used to estimate solutions to ordinary differential equations (ODEs). In this case, we are using Euler's method with n=5 steps over the interval t=[0,1]. The step-by-step process involves finding the step size and using the Euler's method formula to approximate the solution at each step.

Step-by-step explanation:

Euler's method is a numerical method used to estimate solutions to ordinary differential equations (ODEs). To use Euler's method, we need to divide the interval into several subintervals, or steps. In this case, we have n=5 steps over the interval t=[0,1].

Step 1: Find the step size, h, by dividing the interval length by the number of steps: h=1/5=0.2.

Step 2: Use the Euler's method formula to approximate the solution at each step. Start with the initial condition y(0)=3:

yi+1 = yi + h * f(ti, yi)

where ti=0+i*h and f(t,y) represents the derivative of y with respect to t, which is given as y'=y+t^2. Substitute these values into the formula to compute the approximations for each step.