Answer the following items on a clean sheet of paper. Upload photos of your answer sheets showing your solutions. D² f – Df = 0 mm f(0) = Df(0) = 1 2. Solve the initial value problem, with

Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter3: Solving Equation And Problems
Section3.5: Equations With The Variable On Both Sides
Problem 16P
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Answer the following items on a clean sheet of paper. Upload photos of your answer sheets showing your solutions.
D² ƒ – Df = 0 wm
f (0) = Df(0) = 1
2. Solve the initial value problem,
with
Transcribed Image Text:Answer the following items on a clean sheet of paper. Upload photos of your answer sheets showing your solutions. D² ƒ – Df = 0 wm f (0) = Df(0) = 1 2. Solve the initial value problem, with
Solve the initial-value problem:
Df – D²f = 0; f(0) = Df (0) = D²f(0) = 3
Solution:
We apply L to both sides of our ODE
L[D®f] – L[D²f] = L[0] = 0
%3D
By Theorem 1.2,
L[D³f] = s³L[f] – s²f(0) – sDf(0) – D²f(0)
= s³L[f] – 3s² – 3s – 3
L[D²f] = s?L[f] - sf(0) – Df (0)
= s?L[f]– 3s – 3
%D
Substituting into the transformed ODE,
(s°L[f] – 3s? – 3s – 3) – (s²L[f] – 3s – 3) = 0
s³ L[f] – s²L[f] – 3s² = 0
3s2
L[f] =
3
s3 - s2
S- 1
f(x) = L-1
S -
= 3e*
Transcribed Image Text:Solve the initial-value problem: Df – D²f = 0; f(0) = Df (0) = D²f(0) = 3 Solution: We apply L to both sides of our ODE L[D®f] – L[D²f] = L[0] = 0 %3D By Theorem 1.2, L[D³f] = s³L[f] – s²f(0) – sDf(0) – D²f(0) = s³L[f] – 3s² – 3s – 3 L[D²f] = s?L[f] - sf(0) – Df (0) = s?L[f]– 3s – 3 %D Substituting into the transformed ODE, (s°L[f] – 3s? – 3s – 3) – (s²L[f] – 3s – 3) = 0 s³ L[f] – s²L[f] – 3s² = 0 3s2 L[f] = 3 s3 - s2 S- 1 f(x) = L-1 S - = 3e*
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