Consider applying the Trapezoidal method to obtain approximate solutions to an ordinary differential equation of the...

 

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Consider applying the Trapezoidal method to obtain approximate solutions to an ordinary differential equation of the form = f(t,y), y(to) = yo. (1) (5 points) Write out the equation that must be solved at each time step to advance the solution from y, to Yi+1 using the Trapezoidal method. Express this equation as a root- finding problem, and give the formula defining the Newton's method iteration that could be used to determine the solution to this root finding problem. (2) (2 points) At step i, what would you suggest be used as the initial iterate for the Newton's method procedure when it is used to solve the implicit equations that determine y+1 from Yi. (3) (3 points) Consider applying Trapezoidal method to following nonlinear ODE dy dt = (1+t) cos(t/2)cos(y). Give the formula defining Newton's method iteration that could be used to determine Yi+1 from yi. Consider applying the Trapezoidal method to obtain approximate solutions to an ordinary differential equation of the form = f(t,y), y(to) = yo. (1) (5 points) Write out the equation that must be solved at each time step to advance the solution from y, to Yi+1 using the Trapezoidal method. Express this equation as a root- finding problem, and give the formula defining the Newton's method iteration that could be used to determine the solution to this root finding problem. (2) (2 points) At step i, what would you suggest be used as the initial iterate for the Newton's method procedure when it is used to solve the implicit equations that determine y+1 from Yi. (3) (3 points) Consider applying Trapezoidal method to following nonlinear ODE dy dt = (1+t) cos(t/2)cos(y). Give the formula defining Newton's method iteration that could be used to determine Yi+1 from yi.