Euler’s Method Accuracy and Step Size Impact
Consider the second order differential equation \(y” = -y\) with initial conditions \(y(0)=1\) and \(y'(0)=0\). Convert this equation into a system of first order equations and use Euler’s method to approximate \(y(0.1)\) using two different step sizes: \(h=0.1\) and \(h=0.05\). Briefly explain which approximation is more accurate and why.
Part a
Using Euler’s Method, approximate \(y(1)\) for the differential equation \(\frac{dy}{dt} = -y+2\) where \(y(0)=1\) and the step size is 0.5. Recall the Euler update formula: \(y_{n+1}=y_n+h\cdot f(t_n,y_n)\).
Euler’s Method Table Construction
Consider the differential equation \(\frac{dy}{dx}= x+y\) with the initial condition \(y(0)=1\). Use Euler’s method with a step size of \(h=0.5\) to construct a table of iteration steps up to \(x = 1.0\). Your table should list the values of \(x\) and the corresponding approximate values of \(y\) at each step.