### Differential Equation Problem**Initial Value Problem:**\[ y' + 3y = 9t, \quad y(0) = 3. \]**Steps to Solve:**#### a. Laplace TransformApply the Laplace transform to both sides of the differential equation to transform it into an algebraic equation. The Laplace transform of $ y(t) $ is denoted by $ Y(s) $.\[ \frac{9}{s^2} = \]#### b. Solve for $ Y(s) $Solve for $ Y(s) $ using the transformed equation.\[ Y(s) = \mathcal{L} \{ y(t) \} = \frac{(9 + 3s^2)}{(s^2(s+3))} \]#### c. Inverse Laplace TransformApply the inverse Laplace transform to solve for $ y(t) $.\[ y(t) = \]This process involves transforming a differential equation into an algebraic equation using Laplace transforms, solving it, and then transforming it back to find the solution in the time domain.

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Consider the initial value problem y + 3y = 9t, y(0) = 3. 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). 9/s^2 help (formulas) b. Solve your equation for Y(s). Y(s) = L {y(t)} = (9+3s^2/(s^2(s+3)) c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) =

Consider the initial value problem y + 3y = 9t, y(0) = 3. 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). 9/s^2 help (formulas) b. Solve your equation for Y(s). Y(s) = L {y(t)} = (9+3s^2/(s^2(s+3)) c. Take the inverse Laplace transform of both sides of the previous equation to solve for y(t). y(t) =

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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Question

### Differential Equation Problem

**Initial Value Problem:**
\[ y' + 3y = 9t, \quad y(0) = 3. \]

**Steps to Solve:**

#### a. Laplace Transform
Apply the Laplace transform to both sides of the differential equation to transform it into an algebraic equation. The Laplace transform of $ y(t) $ is denoted by $ Y(s) $.

\[ \frac{9}{s^2} = \]

#### b. Solve for $ Y(s) $
Solve for $ Y(s) $ using the transformed equation.

\[ Y(s) = \mathcal{L} \{ y(t) \} = \frac{(9 + 3s^2)}{(s^2(s+3))} \]

#### c. Inverse Laplace Transform
Apply the inverse Laplace transform to solve for $ y(t) $.

\[ y(t) = \]

This process involves transforming a differential equation into an algebraic equation using Laplace transforms, solving it, and then transforming it back to find the solution in the time domain.

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