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.
6) Solve: 4 -2 -2 x'(t) - 2 3 -1 x(t) -1 3 7) Solve using the method of undetermined coefficients. (E -4 -3 2 x'(t) x(t) + 6. -2e* 8) Solve using the method of variations of parameters: 8 x'(t) = x(t) + -1 et %3D te
4. Find the general solution to the equation via the variation of parameters y" + 9y = tan(3t)
Take b = 5 and c = 2
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 a numerical solver and Euler's method to obtain a four-decimal approximation of the indicated value. First use h = 0.1 and then use h = 0.05. y' = ey, y(0) = 0; y(0.5) y(0.5) y(0.5) (h = 0.1) (h = 0.05)
Given that y = sin x is a solution of y(4) + 2y" + 11y" + 2y' + 10y = 0 Find the general solution of the DE and Plot the graph of it.
B1
Use Euler's method to approximate y(1.0). 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 = 1.0 agree rounded off to two decimal places. y' = x² + y² -1, y(0) = 0 The approximate solution values at x = 1.0 begin to agree rounded off to two decimal places between is (Type an integer or decimal rounded to two decimal places as needed.) So, a good approximation of y(1.0) h = 0.001 and h = 0.0001. h = 0.1 and h=0.01. h = 0.01 and h = 0.001.
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!!

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