### 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.

Answered: Image /qna-images/answer/18af0d92-d3ee-47e1-ac56-513dae27e1b6.jpg

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

See similar textbooks

Related questions

Q: Solve the linear differential equation using Fourier transform.

A: The objective of this question is to solve the given linear differential equation using the Fourier…

Q: Sove the differential equation given the condition y(0)=6 dy/dx = (x/y)

Q: Transform the differential equation 4y′+2y=t^(5) y(0)=9 into an algebraic equation by taking…

A: The given differential equation .Using the Laplace transformation to find an algebraic equation and…

Q: write answer as fraction -(1/2)^3

A: We know , (a/b)n = an/bn

Q: Write a differential equation that is a mathematical model of the situation described below:…

A: Use the general equation of a line passing through a point and with given slope to determine the…

Q: nd the solution of the differential equation = Z 3 + sin(z) with i dz

A: Given: dfdz=z-23+sin(z)f(0)=2

Q: Z = 2xy what will be the partial differential coefficient

Q: Explain some of the similarities and some of the differences between using a Laplace transform and…

A: Suppose we choose 1) laplace transfer 2) successful approximation method 1) Laplce transform…

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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Step 1

Step 2

Step 3

Step 4

Trending now

This is a popular solution!

Step by step

Solved in 4 steps with 4 images

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Laplace Transformation$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Determine whether the differential equation is exact. Show work to support your answer. sin(x) dy = y cos(x) dx
Consider the differential equation y" — y' — 6y = et + sint. Find a particular solution. Find the general solution.
Sand is being dumped from a hopper railcar at a rate of 40 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always equal to each other (both are growing larger as more sand is dumped). How fast is the base diameter increasing when the base is 18 feet long? Round your answer to three decimal places. ft/min
Consider the differential equation: dy/dx = y(3 - y) 1) Use the slope field plotter to plot different solutions corresponding to A: (0,-2); B: (0,1); C: (0,3); D: (0,4). You can drag each point to the correct place if they're somewhere else on the plane, or click on the point and redefine. Capture a screenshot showing all the solution curves and insert below. f'(x)=y(3-y) Kin/5 -5 Density Length dy de -8 Step size 0.1 Ymas y (3-9) Input. ✔Solution A Solution B Solution C Solution D DCQ= AC: 2) To what value does y(x) approach, as x → ∞, for the solutions corresponding to B,C, and D? To what value does y(x) approach, as x →∞, for the solution corresponding to A? Why is it different from the others? Be specific and answer in terms of what you know about the slope field and its effect on the solution. - 3) Use wolframalpha.com or something similar to find the particular solution corresponding to each of the points A,B,C,D. How is the solution through Point C different from the other…
The percentage of female high school graduates who enrolled in college is given in the table below. Year 1970 1980 1990 2000 2010 College Enrollment Percentages for Females 25.5 30.3 38.3 45.6 50.5 Let a represent time (in years) since 1970, and let y represent the corresponding percentage of female high school graduates who enrolled in college. Find the cubic regression model y = F(x) = ax³ + bx² + cx + d, = where a is rounded to 6 decimal places, b is rounded to 4 decimal places, c is rounded to 3 decimal places, and d is rounded to 1 decimal place.