Been having trouble with this MATlab assignment, only problem has to be done. After the commands and function has been written the file has to be published uing the command >>publish('euler1.m',pdf). The function is the euler function and the description of the problem is below. Only problem 1 a-c. Thanks again, this problem has had me all week.

MAT 275 Laboratory 3 Numerical Solutions by Euler and Improved Euler Methods scalar equations) In this session we look at basic numerical methods to help us understand the fundamentals of numerical approximations. Our objective is as follows. 1. Implement Euler's method as well as an improved version to numerically solve an IVP 2. Compare the accuracy and efficiency of the methods with methods readily available in MATLAB 3. Apply the methods to specific problens and investigate potential pitfalls of the methods. Instructions: For your lab write-up follow the instructions of LAB 1. Euler's Method To derive Euler's method start from y(to)-o and consider a Taylor expansion at ti-to+h: y(to) + y'(to)(ti-to) + yo + hf(to , y(to)) + y(t) = = For small enough h we get an approximation yi for y(ti) by suppressing the ..., namely Y1 = yo + hf(to,yo) The iteration (L3.1) is repeated to obtain y2y(t2),. .. such that yn+1 tn+1 yn + hf(tn,yn) tn + h = = y(ti) Geometrically, the approximation made is equivalent to replacing the i solution curve by the tangent line at (to, yo). From the figure we have yi-w yo f(to, yo) = f(to, y(to))-y'(to) tan from which (L3.1) follows As an example consider the IVP to t1 y, = 2y=f(t,y) with y(0)= 3. Note that here f does not explicitly depend on t (the ODE is called autonomous), but does implicitly through y y(t). To apply Euler's method we start with the initial condition and select a step size h. Since we are constructing arrays t and y without dimensionalizing them first it is best to clear these names in case they have been used already in the same MATLAB work session. clear t y % no comma between t and y! type help clear for more info ince 1 is simple enough we may use the inline syntaks te amp s Dian MAT 275 Laboratory 3 Numerical Solutions by Euler and Improved Euler Methods scalar equations) In this session we look at basic numerical methods to help us understand the fundamentals of numerical approximations. Our objective is as follows. 1. Implement Euler's method as well as an improved version to numerically solve an IVP 2. Compare the accuracy and efficiency of the methods with methods readily available in MATLAB 3. Apply the methods to specific problens and investigate potential pitfalls of the methods. Instructions: For your lab write-up follow the instructions of LAB 1. Euler's Method To derive Euler's method start from y(to)-o and consider a Taylor expansion at ti-to+h: y(to) + y'(to)(ti-to) + yo + hf(to , y(to)) + y(t) = = For small enough h we get an approximation yi for y(ti) by suppressing the ..., namely Y1 = yo + hf(to,yo) The iteration (L3.1) is repeated to obtain y2y(t2),. .. such that yn+1 tn+1 yn + hf(tn,yn) tn + h = = y(ti) Geometrically, the approximation made is equivalent to replacing the i solution curve by the tangent line at (to, yo). From the figure we have yi-w yo f(to, yo) = f(to, y(to))-y'(to) tan from which (L3.1) follows As an example consider the IVP to t1 y, = 2y=f(t,y) with y(0)= 3. Note that here f does not explicitly depend on t (the ODE is called autonomous), but does implicitly through y y(t). To apply Euler's method we start with the initial condition and select a step size h. Since we are constructing arrays t and y without dimensionalizing them first it is best to clear these names in case they have been used already in the same MATLAB work session. clear t y % no comma between t and y! type help clear for more info ince 1 is simple enough we may use the inline syntaks te amp s Dian