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    Calculus and Analysis

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    Undetermined coefficients

    Use the method of undetermined coefficients to solve the attached differential equation (see attachment)

    Homogeneous Solutions and Differential Equations

    Given that the differential equation y^n + p(x)y' + q(x)y = r(x) attached has three solutions of sin x, cos x and sin 2x. Find yh (yh is the corresponding homogeneous solution). See attachment for better formula representation. See the attached file.

    Wronskian Solution and Differential Equations

    Given that the differential equation y^n + p(x)y' + q(x)y =0 has two solutions x^2 -x and x^3 - x. Use the Wronskian to find p(x). See attachment for better formula representation.

    Wronskians

    Please see the attached file for the fully formatted problems. Determine whether the functions can be Wronskians on -1<x<1 for a pair of solutions to some equation y'' + py' +qy = 0 with p and q continuous. a) W(x) = 6e^4x b) W(x) = x^3 c) W(x) = 0 d) W(x) = (x -1/2)^1/2

    Vector Functions to Partial Derivative

    Attached is more clear 1. Distance from a a point to a curve: Find the shortest distances between the point (1,2,1) and a point on the curve r(t)= (1/t*i)+(lnt(t)*j)+(sqrt(t)*k) 2. Distance from a point to a curve: Find the maxmium distances from the point (1,2,-1) to a point on the curve of intersection of the plane z=(

    Laplace transform and differential equations.

    Find the inverse Laplace transform of (s^3+s^2+2/s) / [s^2(s^2+3s+2)] Using this (or otherwise), Find the solution of the equation y"+3y'+2y = 1-t Find the transform of the following functions: f(t) = (1+t^2)[u(t-1)-u(t-2)] where u(t) is the unit step function. f(t) = sin(t) for 0<t<Pi and f(t)=0 for Pi<t<2*Pi

    Diff. EQ

    Solve for variable y in terms of t W/ given initial condition: dy/dt + 4y = 40sin3t y(0)=6

    Differential Equations

    Solve y in terms of t with initial conditions given. a.) (d^2)y/dt^2+3dy/dt+2y=24e^-4t y(0)=10 y'(0)=5 b.) (d^2)y/dt^2+6dy/dt+9y=0 y(0)=10 y'(0)=0

    Convergent Series Natural Log Series

    Use 1) sum of (x ^ (n +1))/ (n +1 ) converges uniformly on [-1, 0] 2) sum of x ^ n converges uniformly on (-1, 0] 3) sum of x ^ n = 1/(1-x) to show that ln 2 = 1 - 1/2 + 1/3 - 1/4 ...

    Use the Uniqueness Theorem for the Initial Value Problem

    4. For the initial value problem dy/dx = 3y^(2/3), y(2) = 0, (a) does existence uniqueness Theorem 1 imply the existence of a unique solution? Explain. (b) Which of the following functions are solutions to the above differential equation? Explain. (b_1) y(x) = 0 (b_2) y(x) = (x - 2)^3 (b_3) y(x) = (x - alpha)^3, x <

    Derivative Question - Algebraic Function

    This is from a Trig/Calculus course...Explain FULLY: If F(x) = x^4 - 2x^3 + 4x^2 - 9 Note: ^ indicates exponant. Find F prime of x. It will be a derivative. I need every step explained clearly as I have a bet riding on this! I need to be able to show every step in order to win my bet.