### ODE - Method of Variation of Coefficients : (X^2)*y''-2y=Sin(lnx)

ODE - Method of Variation of Coefficients : (X^2)*y''-2y=Sin(lnx)

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ODE - Method of Variation of Coefficients : (X^2)*y''-2y=Sin(lnx)

Integral [(2x^3 - 4x^2 + x + 3) / (x - 1)^2] dx

Integrate both sides of this equation: K(d^2T/dz^2) = w(dT/dz) Derive this equation: A*sin((pi*y)/L)

Let f_n(x) = n^1/2 * x * e^(-n*x^3), for n = 1,2,3... (i) Find the maximum value assumed by f_n in the interval [0,1]. (ii) Find Lim (n -> infinity) of integral from 0 to 1 of (f_n(x))dx. All integrals here are with respect to Lebesgue measure. Please justify every step and claim. e here is the exponential function.

Let {f_n} be a sequence of nonnegative Lebesgue measurable functions on [0,1]. Suppose that: (i) f_n -> f in [0,1] and (ii) integral over [0,1] of f_n =< K for all n and some constant K. Then f is in L^1[0,1] and || f||_1 =< K. All integrals are with respect to Lebesgue measure.

Let a,b be real numbers such that 0 < a < b < infinity. Does the limit lim of ( integral from a to b of n*sin (x^2/n) dx , n is positive integer exist? ( prove or disprove). Find the limit if it exists. Prove all assertions and justify every step. The integral here is with respect of Lebesgue measure.

Compute the quantity limit of ( integral from 0 to 1 e^(-x^2/n) dx) ( the integral here is with respect to Lebesgue measure). Make sure that you verify your manipulations by referring to known theorems.

7. This problem generalizes the factorial function, as in n!=n(n-1)(n-2)...(2)(1), to more general arguments than just the positive integers. (a) Use integration by parts to show that for any positive integer n, the integral with respect to x from 0 to infinity of xne-x is n! (b) Make a clear case that the integral exists

Find exact values for Riemann sums approximating the integral of the function f(x)=x2 on the interval [0,1]. Split up the interval into N equal segments, and find the upper sum (taking the maximum function value in each segment) and the lower sum (taking the minimum function value in each segment). You will need a mysterious loo

Let two long circular cylinders, of diameter D, intersect in such a way that their symmetry axes meet perpendicularly. Let each of these axes be horizontal, and consider the "room" above the plane that contains these axes, common to both cylinders. (In architecture this room is called a "cross vault".) The floor of the cross vau

1. Approximate the integrals using the Trapezoidal rule. a) Integral from -0.5 to 0 x ln(x+1) dx b) Integral from 0.75 to 1.3 ((sin x)2 - 2x sin x +1) dx 2. Find a bound for error in question 1. using the error formula, and compare this to the actual error. 3. Repeat question 1. using Simpson's rule 4. Repeat ques

1) integral(0 to pi/8) sec^2(2x)tan^3(2x) dx 2) integral(dx/xlnx^2) note: only the x is being squared not the whole (lnx)^2

I want to check my answer: Evaluate the following integrals: integral over gamma for (sin z)/z dz, given that gamma(t) = e^(it) , 0=<t=<2pi ( e here is the exponential function) My work: sin z = z - z^3/3! + z^5/5! + ... + (-1)^n (z^(2n-1))/(2n-1)! + ... divide by z we get (sin z)/z = 1 - z^2/3! + z^4/5!

Evaluate the following integrals: (a) integral over gamma of (e^z - e^-z)/(z^n) dz, where n is positive integer and gamma(t) = e^(it), 0 =< t =< 2 pi (b) integral over gamma of (dz/(z^2 + 1) ) where gamma(t) = 2e^(it), 0 =< t =< 2pi ( Hint: expand (z^2 + 1)^-1 by means of partial fractions PLEASE USE POWER SERIE

Compute a triple integral over a specific solid that would be very difficult in rectangular coordinates, but easy in parabolic cylindrical coordinates, u, v, z, where x=(1/2)(u^2-v^2) y=uv z=z You must come up with the solid. Remember that the Jacobian determinant (u^2+v^2) must be used when transforming an integral to this

Let P(z) be polynomial of degree n and let R>0 be sufficiently large so that p never vanishes in { z: |z| >= R}. If gamma(t) = Re^(it), 0 =< t =< 2 pi, show that the integral over gamma p'(z)/p(z) dz = 2 ( pi ) i n.

(a) If f is a nonnegative continuous function on [0,1], then show that integral from 0 to 1 f(x) dx = integral over [0,1] f dx ( that is show that the reimann integral and lebesgue integrals are equal). (b) Prove part (a) for any continuous function.

The problem is the integral of ( (w - 3)^1/2 ) / ( 2 + (w-3)^1/2 ) so its the root of (w-3) divided by the root (w-3) + 2.

(See attached file for full problem description)

(See attached file for full problem description with proper equations) --- 1. For medical purposes the level of sugar was measured in blood (in mg/dl). The samples were taken with 1/2 hour increments, as the following table shows: Initial sample 96 mg/dl After 30 min. 133 mg/dl After 60 min. 142 mg/dl After 90 min.

In a previous problem I posted here: Let f(x) be a positive continuous function on [0,1/2], f(x) =< 1/2. Let A = { (x,y) : 0 =< x = 1/2, 0=<y=< f(x)} Prove that; m*(A) = integral from 0 to 1/2 of f(x)dx. Now knowing that the above is true, I want to show that the integral is lebesgue measurable, that is, the area un

Let X be an uncountable set, let m be the collection of all sets E in X such that either E or E^c is at most countable, and define M(E) = 0 in the first case, and M(E) = 1 in the second case. ( m here is sigma algebra in X). The Questions is : Describe the integrals of the corresponding measurable functions.

1) the indefinite integral of du/(u(a+bu))=(1/a) ln |u/(a+bu)| + c in words the indefinite integral of du over the quantity of u times the quantity of a plus b times u... 2) the indefinite integral of du/(u((a+bu)^2))= 1/(a(a+bu))= 1/(a^2) ln | (a+bu)/u| +c the indefinite integral of du over the quantity of u times t

1) The integral of dx divided by the the quantity of e^-x+1 also known as the indefinite integral of dx/((e^-x)+1) 2) Solve the indefinite integral of the quantity of 1 minus e^x all over the quantity of 1 plus e^x with respect to x... also know as the indefinite integral of (1-e^x)/(1+e^x) dx

1) indefinite integral of dx / ((x^2)(sqrt[a^2-x^2])) with respects to x In words it is the integral of the derivative of x all over the quantity of x squared times the square root of a squared minus x squared 2) indefinite integral of dx / ((e^x)(sqrt[4+e^2x])) with respects to x in words it is the integral of the der

See attached pdf file.

I need help on how to work out the solution to a function using the 'D-contour' (see attached file).

Find the integral of f(x) = x tan^-1 x/(1+x^2)^2

(See attached file for full problem description with proper symbols) --- Let and for (a) Use integration by parts to show that in for . Deduce that for (b) Compute for and verify that ---

1).If A is a subset of B, A,B in m ( measurable sets) then show that integral (region A) s dM =< integral ( region B) s dM Where s here is a simple non-negative measurable function. ( Please don't confuse this with bounded measurable functions, I need the proof for SIMPLE functions). 2). If E are measurable, X_E is the c