y'=x²y², y(¹)=1a) Calculate the Euler method approximationsto the solution to the initial value problemx = 1.2, 1.4, 1.6, and 1.8. (Round your answersto 5 decimal places,atb) Find the exact solution to the initial valueproblem.c) Construct a chart to compare the resultsin part (a) with the exact solution atthe given x-values.

The given differential equation is y'(x)=x2y2 with the initial condition y(1) =1. Here, f(x, y) =…

Answered: y'=x²y², y(¹)=1 a) Calculate the Euler…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Use Euler's method to approximate y(1.3). Start with step size h = 0.1, and then use successively smaller step sizes (h=0.01, 0.001, 0.0001, etc.) until successive approximate solution values at x = 1.3 agree rounded off to two decimal places. y' = x² + y² -1, y(0) = 0 The approximate solution values at x = 1.3 begin to agree rounded off to two decimal places between h = 0.01 and h = 0.001. So, a good approximation of y(1.3) is. d to two decimal places as needed.) h = 0.01 and h = 0.001. h = 0.1 and h=0.01. h = 0.001 and h = 0.0001.
4. Find the general solution to the equation via the variation of parameters y" + 9y = tan(3t)
The following second-order equations explicitly contain y. Solve using the substitution v = y' (a) xy''+4y'=18x2
Use Euler's method to approximate y(1.8). Start with step size h=0.1, and then use successively smaller step sizes (h=0.01, 0.001, 0.0001, etc.) until successive approximate solution values at x = 1.8 agree rounded off to two decimal places. y'=x² + y²-2. y(0) = 0 The approximate solution values at x = 1.8 begin to agree rounded off to two decimal places between (Type an integer or decimal rounded to two decimal places as needed.) ▾ So, a good approximation of y(1.8) is.
Use Euler's method to approximate y(1.9). Start with step size h = 0.1, and then use successively smaller step sizes (h = 0.01, 0.001, 0.0001, etc.) until successive approximate solution values at x = 1.9 agree rounded off to two decimal places. y' =x? +y? - 2, y(0) = 0 The approximate solution values at x = 1.9 begin to agree rounded off to two decimal places between h= 0.001 andh= 0.0001. So, a good approximation of y(1.9) is 0.11.
Y 11) Given y'= xy² + and y(1) = 1. X a) Use the Euler's and RK4 Methods to find a four decimal approximation of y(1.5) using h = 0.1. b) Find the Exact solution analytically. c) Provide a table of values for all three (Exact, Euler, and RK4) methods leading to y(1.5) and plot them on the same axis, as discussed in class!!
Apply the improved Euler method to approximate the solution on the interval [0, 0.5] with step size h= 0.1. Construct a table showing values of the approximate solution and the actual solution at the points x 0.1, 0.2, 0.3, 0.4, 0.5. y =y-x-3, y(0) = 1; y(x) = 4 +x-3 e* Complete the table below. (Round to four decimal places as needed.) 0.1 0.2 0.3 0.4 0.5 Actual, y (Xn) Improved Euler, yn

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