Question

Use the Laplace transform to solve the initial value problem. y'' + 4y' + 3y = 1 - u(t - 2) - u(t - 4) + u(t - 6), y(0) = 0, y'(0) = 0

          Use the Laplace transform to solve the initial value problem.
y'' + 4y' + 3y = 1 - u(t - 2) - u(t - 4) + u(t - 6), y(0) = 0, y'(0) = 0
        
Use the Laplace transform to solve the initial value problem.
y” + 4y' + 3y = 1 - u(t - 2) - u(t - 4) + u(t - 6), y(0) = 0, y'(0) = 0

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Calculus: Early Transcendentals
Calculus: Early Transcendentals
James Stewart 8th Edition
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Use the Laplace transform to solve the initial value problem. y" + 4y' + 3y = 1 - u(t - 2) - u(t - 4) + u(t - 6), y(0) = 0, y'(0) = 0
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Transcript

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00:01 Hello everyone, we need to solve the given initial value problem using laplas transform.
00:06 So let us take laplice transform on both sides of the given initial value problem.
00:13 So that will be laplice of y double dash plus 4y dash plus 3y is equal to laplice transform of 1 minus u of t minus 2 minus u of t minus 4 plus u of t minus 6 so here let us use the formula the lapless transform of 1 is nothing but 1 by s and the lapless transform of u of t minus a is equal to e power minus as by s we can simplify the obtained expression as laplas transform of y double dash, plus 4 times laplice transform of y dash, plus 3 times laplice transform of y is equal to laplice transform of 1, minus laplace transform of u of t minus 2, minus laplace transform of u of t minus 4, lapless transform of u of t minus 6.
01:38 So on further simplifying this, we would get s -squire y of s minus s -y of 0 minus y -dash of 0 plus 4 times s -y -of -s minus y of 0 plus 3 -y -of -s is equal to 1 by s minus e power minus 2 s by s minus e power minus 4 s by f plus e power minus 6 s by s on further simplifying this we would obtain s square plus 4 s plus 3 into y of s is equal to 1 by s minus e power minus 2 s by s by minus e power minus 4 s by s plus e power minus 6 s by s so the left hand side can be factized as s plus 1 into s plus 3 into y of s is equal to 1 by s minus e power minus 2 s by s minus e power minus 2 s by s minus e power minus 4 s by s plus e power minus 6 s by s.
03:19 So this can be further simplified as y of s is equal to 1 minus e power minus 2 s minus e power minus 4 s plus e power minus 6 s by s into s plus 1 into s plus 3...
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