Answered: where f(t)= 2: y'−y=f(t), y(0) = 1…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Related questions

Question

Solve the initial-value problem
y'-y = f(t), y(0)=1
where f(t)={(2;,0<=t<=1),(-1;,t>=1)

Using the Laplace transform

where f(t)
=
√2;
-1;
y'-y = f(t), y(0) = 1
0 < t < 1
t>1

Expert Solution

Step 1: We write properties of Laplace transform. By

(.) Given initial value problem is,

$y apostrophe minus y space equals space f left parenthesis t right parenthesis space space space space semicolon space space space space y left parenthesis 0 right parenthesis equals 1$

where $f left parenthesis t right parenthesis space equals space space open curly brackets table row cell 2 space space space space space space semicolon space space 0 less or equal than t less or equal than 1 end cell row cell negative 1 space space space semicolon space space space t greater or equal than 1 end cell end table close$

(.) Some properties of Laplace transform:

$left parenthesis asterisk times right parenthesis space space space L open curly brackets y apostrophe left parenthesis t right parenthesis close curly brackets space equals space s L open curly brackets y left parenthesis t right parenthesis close curly brackets minus y left parenthesis 0 right parenthesis left parenthesis asterisk times right parenthesis space space space space L open curly brackets e to the power of a t end exponent close curly brackets space equals space space fraction numerator 1 over denominator left parenthesis s minus a right parenthesis end fraction open parentheses asterisk times close parentheses space space space space L open curly brackets u left parenthesis t minus a right parenthesis space close curly brackets space equals space space e to the power of negative a s end exponent over s$

(.) Unit step function: A unit step function $u left parenthesis t minus a right parenthesis$ is defined as,

$u left parenthesis t minus a right parenthesis space equals space space open curly brackets table row cell 0 space space space semicolon space space space t less than a end cell row cell 1 space space space semicolon space space space t greater or equal than a end cell end table close$