Numerical Methods Assignment: Euler, Modified Euler and Runge-Kutta Methods, Assignments of Engineering Mathematics

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Extra Practice Assignments: Solve the following by manual calculation and R programming. 1. 10. 1 12. Solve the equation 2 =1+xy, x9 = 0, yo = 1to find yatx = O.landx = 0.2 using modified Euler’s method taking h = 0.1 . Use Euler’s method and modified Euler’s method to solve the equation x =x*+y subject to the conditions atx = 0, y = landtabulatey for x = 0, 0.1and0.5 , a dl . ops . Use Euler’s method to solve the equation 2 =1+xy subject to the conditions at x = 0, y = land tabulate y for x = 0, 0.1and0.5 By Euler’s method, the value of y corresponding to x = 0.2 is, given that x =xty, with y(0) = 1 taking h = 0.2 Using Euler’s method, find an approximate value of y corresponding to x = 1.4, given Pax yandy=1latx=1 Using Modified Euler’s method, find y(0.4) given y’ =x+sin(y), y(0) =1. Take h = 0.2 Using Modified Euler’s method, find y(0.4) given y’=x+ |vyl, y(0) =1. Take h = 0.2 Apply Runge-Kutta method to find an appropriate value of y atx = 0.2 in steps in 0.1, if Baye +x given that y = 1,whenx = 0 Using Runge-Kutta method of fourth order, solve x =xy, y(1) = 2at x=1.2 with h = 0.2 Using Runge-Kutta method of fourth order, solve & =-y’?+x, y(0) = 1, to find y(0.4) withh = 0.1 . Given the differential equation 2 = “ with y(2) = 2. Estimate y(2.5) using the Rung- Kutta method with h = 0.5 Using Runge-Kutta method of fourth order, solve 2 =xy, y(1)= 2at x = 1.2withh = 0.2 . Use Runge — Kutta method of fourth order to find y(0.2) with h = 0.1 for the initial a value problem = = yxty, y(0)=1. 14. Using Runge-Kutta method of order 4, find y(0.2) for the equation Bo¥* yo) =1. dx y+x 15. Using Runge-Kutta method of order 4, find y(1.2) for the equation dy _ 2xyte* y(1) =0 dx x?+xe* ’ 16. Using Runge-Kutta method of order 4, find y(0.4) given y’=x+ |vy|-y(o) =1 take h = 0.2