Chapter 15: Problem 22

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+2 y^{\prime}+5 y=10 \cos t, \quad y_{0}=2, \quad y_{0}^{\prime}=1 $$

Short Answer

Expert verified

Solve the differential equation using Laplace transforms, apply initial conditions, solve for $Y(s)$, use partial fraction decomposition, and take the inverse Laplace transform.

Step by step solution

Apply the Laplace transform

Apply the Laplace transform to both sides of the differential equation. Recall the Laplace transform properties for derivatives: \[ L\{y''\}=s^2Y(s)-sy(0)-y'(0) \] \[ L\{y'\}=sY(s)-y(0) \] So the transformed equation is: \[ s^2Y(s) - sy(0) - y'(0) + 2(sY(s) - y(0)) + 5Y(s) = L\{10 \cos t\} \]

Substitute initial conditions and simplify

Substitute the initial conditions $ y(0) = 2 $ and $ y'(0) = 1 $ into the equation: \[ s^2Y(s) - 2s - 1 + 2(sY(s) - 2) + 5Y(s) = \frac{10s}{s^2 + 1} \] Simplify the equation: \[ (s^2 + 2s + 5)Y(s) - 2s - 5 = \frac{10s}{s^2 + 1} \]

Solve for $Y(s)$

Isolate $Y(s)$ by combining like terms: \[ (s^2 + 2s + 5)Y(s) = \frac{10s}{s^2 + 1} + 2s + 5 \] \[ Y(s) = \frac{10s}{(s^2 + 1)(s^2 + 2s + 5)} + \frac{2s + 5}{s^2 + 2s + 5} \]

Perform partial fraction decomposition

To simplify $ Y(s) $, use partial fraction decomposition. The first term can be decomposed as: \[ \frac{10s}{(s^2 + 1)(s^2 + 2s + 5)} = \frac{As + B}{s^2 + 1} + \frac{Cs + D}{s^2 + 2s + 5} \] Similarly, decompose the second term if needed, and solve for constants by equating coefficients.

Take the inverse Laplace transform

Using the simplified form of $ Y(s) $ from partial fraction decomposition, take the inverse Laplace transform to find $ y(t) $. Use known inverse transforms for each term, such as: \[ L^{-1} \{ \frac{s}{s^2 + a^2} \} = \cos(at) \] \[ L^{-1} \{ \frac{a}{s^2 + a^2} \} = \sin(at) \] Combine the results to express $ y(t) $ in its final form.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Differential Equations

Differential equations involve functions and their derivatives. They're essential for modeling various phenomena in science and engineering.
For example, in the exercise, we're given a second-order differential equation: $ y'' + 2y' + 5y = 10 \cos{t} $.
Solving such equations often involves finding the unknown function (here, $ y(t) $) that satisfies the equation. Differential equations can be solved using various methods, and the choice of method depends on the type and complexity of the equation.
In this exercise, we use Laplace transforms.

Initial Conditions

Initial conditions specify the state of the system at the beginning of the observation.
For differential equations, these conditions help us find a unique solution. In our problem, the initial conditions given are: $ y(0) = 2 $ and $ y'(0) = 1 $.
These conditions allow us to determine the particular constants when we transform and solve the equation.
Without initial conditions, the solution to a differential equation might not be unique. They help to tailor the generalized solution to fit specific scenarios, which is critical for real-world applications.

Partial Fraction Decomposition

Partial fraction decomposition is a method to simplify complex rational expressions.
This process involves breaking down a fraction into simpler parts to make the inverse Laplace transform easier.
In our example, after transforming the differential equation and isolating $Y(s)$, we get: \[ Y(s) = \frac{10s}{(s^2 + 1)(s^2 + 2s + 5)} + \frac{2s + 5}{s^2 + 2s + 5} \].
To simplify, we decompose the first term as: \[ \frac{10s}{(s^2 + 1)(s^2 + 2s + 5)} = \frac{As + B}{s^2 + 1} + \frac{Cs + D}{s^2 + 2s + 5} \].
Solving for constants $A, B, C,$ and $D$ helps us handle the terms individually, making it easier to find the inverse Laplace transforms.

Inverse Laplace Transform

The inverse Laplace transform converts a function from the s-domain (Laplace domain) back to the time domain.
This final step gives us the solution to the original differential equation.
After partial fraction decomposition, we take the inverse transform of each term.
For instance: \[ L^{-1} \{ \frac{s}{s^2 + a^2} \} = \cos(at) \] and \[ L^{-1} \{ \frac{a}{s^2 + a^2} \} = \sin(at) \].
By combining the inverse transforms of all terms, we arrive at the final solution for $ y(t) $.
In our problem, these standard inverse transforms help us reconstruct the time-domain function that satisfies our differential equation and initial conditions.

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}+2 y^{\prime}+5 y=10 \cos t, \quad y_{0}=2, \quad y_{0}^{\prime}=1 $$

Short Answer

Step by step solution

Apply the Laplace transform

Substitute initial conditions and simplify

Solve for \(Y(s)\)

Perform partial fraction decomposition

Take the inverse Laplace transform

Key Concepts

Differential Equations

Initial Conditions

Partial Fraction Decomposition

Inverse Laplace Transform

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