let a be a non zero constant and consider: y''+(1/t)y'=a a. show that 1 and ln(t)...

let a be a non zero constant and consider: y''+(1/t)y'=a

a. show that 1 and ln(t) are linear independent solutions of the corresponding homogenous equation

b. using variation of parameters find the particular solution to the non homogenous equation

c. express the solution to the non homogenous equation in terms of a. and b.

d.since y itself does not appear in the equation, the substitution w=y' can be used to reduce the equation to a linear 1st order equation. use this substitution to solve for w directly using a 1st order technique and verify that the two techniques produce the same answer

Expert Solution

Colby Messinger answered 2 years ago

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if ...

1. Let X and Y be non-linear spaces and T : X -->Y. Prove that if T is One-to-one then T-1 exist on R(T) and T-1 : R(T) à X is also a linear map. 2. Let X, Y and Z be linear spaces over the scalar field F, and let T1 ϵ B (X, Y) and T2 ϵ B (Y, Z). let T1T2(x) = T2(T1x) ∀ x ϵ X. (i) Prove that T1T2 ϵ B (X,Y) is also a...

1. Consider the initial value problem y′ =1+y/t, y(1)=3 for1≤t≤2. • Show that y(t) = t...

1. Consider the initial value problem y′ =1+y/t, y(1)=3 for1≤t≤2. • Show that y(t) = t ln t + 3t is the solution to the initial value problem. • Write a program that implements Euler’s method and the 4th order Runke-Kutta method for the above initial value problem. Use your program to solve with h = 0.1 for Euler’s and h = 0.2 for R-K. • Include a printout of your code and a printout of the results at each...

Let y(t) = (1 + t)^2 solution of the differential equation y´´ (t) + p (t) y´...

Let y(t) = (1 + t)^2 solution of the differential equation y´´ (t) + p (t) y´ (t) + q (t) y (t) = 0 (*) If the Wronskian of two solutions of (*) equals three. (a) ffind p(t) and q(t) (b) Solve y´´ (t) + p (t) y´ (t) + q (t) y (t) = 1 + t

Let x, y, z be (non-zero) vectors and suppose w = 12x + 18y + 4z...

Let x, y, z be (non-zero) vectors and suppose w = 12x + 18y + 4z If z = − 2x − 3y, then w = 4x + 6y Using the calculation above, mark the statements below that must be true. A. Span(w, x, y) = Span(w, y) B. Span(x, y, z) = Span(w, z) C. Span(w, x, z) = Span(x, y) D. Span(w, z) = Span(y, z) E. Span(x, z) = Span(x, y, z)

Q1: Let x = [ x(t) y(t) ] and consider the system of ODEs x' =...

Q1: Let x = [ x(t) y(t) ] and consider the system of ODEs x' = [5/2, 3; −3/4 ,−1/2] x. (1) 1.1 Solve the initial value problem subject to x(0) = 1, y(0) = 1.

The temperature T at (x,y,z) in the 3D space is given by T(x,y,z) = ln(1+x2y2+z2). a)...

The temperature T at (x,y,z) in the 3D space is given by T(x,y,z) = ln(1+x2y2+z2). a) Find rate of change of T at the point P(1,-1,-1) in the direction of Q(2,0,0)? b) In which direction from P(1,-1,-1) does the temperature T increase most rapidly? c) What is the maximum rate of change of T at P(1,-1,-1)?

Let g(t) = 5t − 3 ln(t 2 ) − 4(t − 2)2 , 0.1 ≤...

Let g(t) = 5t − 3 ln(t 2 ) − 4(t − 2)2 , 0.1 ≤ t ≤ 3. a. Find the absolute maximum and minimum of g(t). b. On what intervals is g(t) concave up? Concave down?

Consider the IVPs: (A) y'+2y = 1, 0<t<1 , y(0)=2. (B) y' = y(1-y), 0<t<1 ,...

Consider the IVPs: (A) y'+2y = 1, 0<t<1 , y(0)=2. (B) y' = y(1-y), 0<t<1 , y(0)=1/2. 1. For each one, do the following: a. Find the exact solution y(t) and evaluate it at t=1. b. Apply Euler's method with Δt=1/4 to find Y4 ≈ y(1). Make a table of tn, Yn for n=0,1,2,3,4. c. Find the error at t=1. 2. Euler's method is obtained by approximating y'(tn) by a forward finite difference. Use the backward difference approximation to y'(tn+1)...

Find an equation of the tangent to the curve x = 2 + ln t, y...

Find an equation of the tangent to the curve x = 2 + ln t, y = t2 + 4 at the point (2, 5) by two methods. (a) without eliminating the parameter (b) by first eliminating the parameter

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (?...

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (? − 1)? + ?? is ?(??+1). 7. Arrange the functions ?10, 10?, ? log ? , (log ?)3, ?5 + ?3 + ?2, and ?! in a list so that each function is big-O of the next function. 8. Give a big-O estimate for the function ?(?)=(?3 +?2log?)(log?+1)+(5log?+10)(?3 +1). For the function g in your estimate f(n) is O(g(n)), use a simple function g...

Question