Consider the initial value problemy + 6y=110if 0 < t < 1if 1 < t < 7if 7 < t < ∞0,y(0) = 4.a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation.Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you getto part (b) below).=b. Solve your equation for Y(s).Y(s) = L{y(t)} =c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t).y(t) =help (formulas)

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Answered: Consider the initial value problem 0…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

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Use the Laplace transform to solve the following initial value problem: y" + 3y = 0 y(0) = 4, y(0) = 1 a. Using Y for the Laplace transform of y(t), i.e., Y = L{y(t)}, find the equation you get by taking the Laplace transform of the differential equation = 0 b. Now solve for Y(s) = c. Write the above answer in its partial fraction decomposition, Y(s) = where a < b Y(s) = [ d. Now by inverting the transform, find y(t) = ■. A B s+a s+b Il+ +
Provide the General Solution of each differential equations. Box your Final Answer and show your solutions:
Consider the following initial-value problem. y' + y = t sin(t), y(0) = 0 Take the Laplace transform of the differential equation and solve for L{y}. (Write your answer as a function of s.) L{y} = Use the Laplace transform to solve the given initial-value problem. Use the table of Laplace transforms as needed. y(t) =
Consider the initial value problem + 2y = 12t, y(0) = 6. a. Take the Laplace transform of both sides of the given differential equation to create the corresponding algebraic equation. Denote the Laplace transform of y(t) by Y(s). Do not move any terms from one side of the equation to the other (until you get to part (b) below). help (formulas) b. Solve your equation for Y(s). Y(s) = L{y(t)} = c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) =
Recall that the differential equation for the instantaneous charge g(t) on the capacitor in an LRC-series circuit is * Rdg dt R9 +9 = E(t). dt2 See this excerpt about LRC-series circuits. Use the Laplace transform to find q(t) when L = 1 h, R = 20 N, C = 0.005 f, E(t) = 130 V, t > 0, q(0) = 0, and i(0) = 0. q(t) = What is the current i(t)? i(t) =
Recall that the differential equation for the instantaneous charge q(t) on the capacitor in an LRC-series circuit is MY NOTES ASK YOUR TEACHER + Rda = E(t). + dt 2 dt C See this excerpt about LRC-series circuits. Use the Laplace transform to find q(t) when L=1 h, R = 20 N, C = 0.005 f, E(t) = 160 V, t > 0, q(0) = 0, and i(0) = 0. q(t) = What is the current i(t)? i(t)
13) The integer n = a) If n = 3 (mod 9), then what is x ? b) If n = 3 (mod 11), then what is x ? Hint for #13: 12700140x3243243 is missing the digit x. Consider the number 1234. By writing each digit in scientific notation, we have 1234 1 x 10³ +2× 10² +3 × 10¹ + 4 × 10⁰ Then notice that 10 = 1 (mod 9). When we reduce mod 9, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x 1³ + 2 x 1² + 3 × 1¹ + 4 × 1⁰ = 1+ 2+ 3+ 4 = 10 = 1 (mod 9). Since 10 = -1 (mod 11), when we reduce mod 11, we obtain 1234 1 x 10³ +2× 10² +3 x 10¹ + 4 x 10⁰ = = 1x (-1)³ +2× (-1)² +3 × (-1)¹ +4× (-1)⁰ = -1 + 2-3 + 4 = 2 (mod 11).
3. Use the Laplace transform to solve the given differential equation. x' - 2x - 4y = 4, y' + 2x + 2y = t, x(0) = 3, y(0) = 1. %3D

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Consider the initial value problem 0 y+6y= 11 0 if 0 < t < 1 if1