cue928 said:Homework Statement
y' = y - x - 1, y(0) = 1, h = .25
Homework Equations
The Attempt at a Solution
y1 = 1+(.25)*(1-0-1) = 1
y2 = 1+(.25)*(1-1-1) = .75
This is not what the book has, but it is organized weird to me. It asks for EM twice, first w/ step size h=.25, then h=0.1. But the way the answers are listed in the back do not correspond with this so I am not sure what they are getting. Curious if folks agree w/ me here on y1, y2. Thanks in advance.
Euler's method is a numerical method used to approximate the solution of a first-order ordinary differential equation (ODE). It involves dividing the interval of the ODE into smaller intervals and using the slope at a given point to estimate the next point on the curve.
Euler's method is used when an analytical solution to an ODE is not available or when it is difficult to solve analytically. It is also commonly used in computer programs to approximate the solution of ODEs.
Euler's method can introduce significant errors in the approximation, especially when the step size is large. It also cannot handle stiff ODEs, where the solution changes rapidly, and the step size needs to be very small to get an accurate approximation.
Euler's method is implemented by first choosing a starting point and a step size. Then, using the formula y_n+1 = y_n + h*f(x_n,y_n), where h is the step size and f(x_n,y_n) is the slope at the current point, the next point on the curve is estimated. This process is repeated until the desired interval is covered.
Some alternative methods to Euler's method include the improved Euler's method, the Runge-Kutta method, and the Adams-Bashforth method. These methods use more sophisticated calculations to provide more accurate approximations of the solution to ODEs.