2. Given two initial value problems, у" — р(г)у +q()у +r(x) with a <I<b,y(a) — с,1 (а) —0 (1) and у" — р(...
Given a second order linear homogeneous differential equation а2(х)у" + а (х)У + аo(х)у — 0 we know that a fundamental set for this ODE consists of a pair linearly independent solutions yı, V2. But there are times when only one function, call it y, is available and we would like to find a second linearly independent solution. We can find y2 using the method of reduction of order. First, under the necessary assumption the a2(x) F 0 we rewrite...
9. Let x,y > 0 be real numbers and q, r E Q. Prove the following: (а) 29 > 0. 2"а" and (29)" (b) x7+r (с) г а — 1/29. 0, then x> y if and only if r4 > y (d) If q (e) For 1, r4 > x" if and only if q > r. For x < 1, x4 > x* if and only if q < r.
a) Write the linear initial value problem with constant matrix A, У — Ау, у(0) %3D Уо, = integral equation. as an b) Apply the method of successive substitution and find the nth iterate yn(x). Evaluate, lim y(x) n oo to get the solution to the IVP.
Consider the following initial value problem у(0) — 0. у%3D х+ у, (i) Solve the differential equation above in tabular form with h= 0.2 to approximate the solution at x=1 by using Euler's method. Give your answer accurate to 4 decimal places. Given the exact solution of the differential equation above is y= e-x-1. Calculate (ii) all the error and percentage of relative error between the exact and the approximate y values for each of values in (i) 0.2 0.4...
Find the solution of the initial value problem [y(0)=-1, у"(0)=-3, у "(0)=0, у"(0)-0] a) y--3-х -2x-2x f None of the above.
Solve using MATLAB (1) Consider the difference equation given follows as у(п) — — 0.9у(п - 1) — 0.81у(n — 2) + 1.62г(п) + 1.82(п — 1) + 2ar (п — 2), п > 0. Find and plot x(n) and y(n) and hence find the energy of y(n) for the following cases (а) 2(п) — 8(п — 2) (b) 2(п) — и(п) 6(п — 3) _ (с) г(п) — и(п)u(-п + 23) _ (d) x(n)r(n) - r(n - 25), where...
(л +у)? )1 (а) Find or show that it doesn't exist lim (x,y)-(0,0) 2y2 (b) At what points in R2 is the function (x + y)2 if (r, y)(0,0), f(x,y) otherwise brief explanation continuous? Give a (л +у)? )1 (а) Find or show that it doesn't exist lim (x,y)-(0,0) 2y2 (b) At what points in R2 is the function (x + y)2 if (r, y)(0,0), f(x,y) otherwise brief explanation continuous? Give a
Solve the initial-value problem. y" (0) =-1 y(0) = 2, y'(0) = 2, у",-2y" + y,-xe* + 5,
Discretization, ODE solving, condition number. Consider the differential equation 5y"(x) - 2y'(x) +10y(x)0 on the interval x E [0,10] with boundary conditions y(0)2 and y (10) 3 we set up a finite difference scheme as follows. Divide [0,10] into N-10 sub-intervals, i.e. {xo, X1, [0,1,. 10. Denote xi Xo + ih (here, h- 1) and yi E y(x). Approximate the derivatives as follows X10- 2h we have the following equations representing the ODE at each point Xi ,i = 1,...
Solve the given initial-value problems (a) y, +(Ian x)y=cos2 x, y(0)--I [yssin x cos x-cos x]