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Use the Laplace transform to solve the given initial-value problem. y' + 6y=e³, y(0) = 2 Step 1 To use the Laplace transform to solve the given initial value problem, we first take the transform of each member of the differential equation dy +6y=e³t. The strategy is that the new equation can be solved for L{y} algebraically. Once solved, transforming back to an equation for y gives the solution we need to the original differential equation. + L{by} = {e³t} We now recall the following, where a and b are constants. • By Theorem 7.2.2.: . Using linearity of L: By Theorem 7.1.1.: Applying these gives the following result. sL{y} -y(0) + Lay} = a£{y} L{eot} = 1 s-b = s£{y} = y(0) L{y} = S-

Use the Laplace transform to solve the given initial-value problem. y' + 6y=e³, y(0) = 2 Step 1 To use the Laplace transform to solve the given initial value problem, we first take the transform of each member of the differential equation dy +6y=e³t. The strategy is that the new equation can be solved for L{y} algebraically. Once solved, transforming back to an equation for y gives the solution we need to the original differential equation. + L{by} = {e³t} We now recall the following, where a and b are constants. • By Theorem 7.2.2.: . Using linearity of L: By Theorem 7.1.1.: Applying these gives the following result. sL{y} -y(0) + Lay} = a£{y} L{eot} = 1 s-b = s£{y} = y(0) L{y} = S-

Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Use the Laplace transform to solve the given initial-value problem. +6y=e³t, y(0) = 2 Step 1 To use the Laplace transform to solve the given initial value problem, we first take the transform of each member of the differential equation +6y=e³t. The strategy is that the new equation can be solved for {y} algebraically. Once solved, transforming back to an equation for y gives the solution we need to the original differential equation. d't + L{6y} = £{e³t} We now recall the following, where a and b are constants. dy = s£{y} − y(0) L{ay} = aL{y} £{eot} = • By Theorem 7.2.2.: • Using linearity of L: • By Theorem 7.1.1.: Applying these gives the following result. s{y}y(0) + ]]£{y} =] s-b
Transcribed Image Text:Use the Laplace transform to solve the given initial-value problem. +6y=e³t, y(0) = 2 Step 1 To use the Laplace transform to solve the given initial value problem, we first take the transform of each member of the differential equation +6y=e³t. The strategy is that the new equation can be solved for {y} algebraically. Once solved, transforming back to an equation for y gives the solution we need to the original differential equation. d't + L{6y} = £{e³t} We now recall the following, where a and b are constants. dy = s£{y} − y(0) L{ay} = aL{y} £{eot} = • By Theorem 7.2.2.: • Using linearity of L: • By Theorem 7.1.1.: Applying these gives the following result. s{y}y(0) + ]]£{y} =] s-b
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