3) Consider the initial value problem ' y' + 8y = 0, y(0) = -4, y (0) = 16 What is the solution of this initial value problem? A) y = -4t - 2e‍8t D) y = -4 + 2e-8t B) y = -2 + 2e8t C) y = -2 -2e-8t E) y = -4+ 2e8t F) y = -2t-2e-8t

Show that the equation x³ + 3y3 = 923 has no solution in integers, other than x = y = z = 0.

3) Personal taxes Suppose that the percent of total personal income that is used to pay personal taxes is given by y = 0.034x² – 0.044x + 12.642 where x is the number of years past 1990 (Source: Bureau of Economic Analysis, U.S. Department of Commerce). (a) Verify that x = 7 is a solution of 14 = 0.034x² – 0.044x + 12.642. (b) Find the year or years when the percent of total per- sonal income used to pay personal taxes is 14.

Find a particular solution to -14.5e-31 y" +6y +9y = 12 +1 Yp

Nitin.jgfff

10. Please provide a solutuion for this equation thankyou

Find the general solution to y(4) – 6y " +8y" = 0. In your answer, use c1, C2, C3 and C4 1o denote arbitrary constants and x the independent variable. Enter c as "c1", c2 as c2", etc. Enter the solution as an equation y =?. y=(c1)+(c1)^x+(c3)e^(3x)+(c4)e^(5x) help (equations)

Ex. 61. Solve 2zx-px²-2qxy+ pq=0.

Y₁ = 1+ a3x³ + A6㺠+... Y₂ = x + b₁x¹ +b7x7 +... Enter the first few coefficients: Az = a6 = b₁ b7 ) Find two linearly independent solutions of y" + 6xy = = - 0 of the form

1. The first three terms of a GP are 1, x, y and the first 3 terms of an AP are 1, x, - y. Prove that x^2 + 2x - 1 = 0, and hence find y, given that x is positive. 2. Show that for an AP with first term a, common difference d and sum S, the number of terms n, must satisfy the quadratic equation given below. Find n when a = 3, d = ½, and S = 2828. Show derivation in your solutions and write the value of n in this form that will satisfy the premises. (photo will be attached ) 3. The sum of three numbers in an A.P. is 12 and the sum of their cubes is 408. Find the terms. 4. A geometric progression has positive terms. The sum of the first six terms is nine times the sum of the first three terms. The seventh term is 320. Find the common ratio and the first term. Find the smallest value of n such that the sum to first n terms of the progression exceeds 10^6 . (Show complete solutions and write answers in this form as: r=___, a=____, n=___. e.g.: r=4, a=7, n=20)