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Chapter 15: Problem 30

Solve the following sets of equations by the Laplace transform method. $$ \begin{array}{ll} y^{\prime}+2 z=1 & y_{0}=0 \\ 2 y-z^{\prime}=2 t & z_{0}=1 \end{array} $$

Short Answer

Expert verified
y(t) = \frac{1}{2}(1 - \cos(2t)), z(t) = \frac{1}{2}(1 + \sin(2t))

Step by step solution

01

Apply Laplace Transform

Apply the Laplace Transform to both equations. Recall that the Laplace Transform of a derivative is given by \[ L\{y'\} = sY(s) - y(0) \] and \[ L\{z'\} = sZ(s) - z(0) \]. Given initial conditions \( y(0) = 0 \) and \( z(0) = 1 \), the transformed system is: \[ sY(s)+2Z(s)=\frac{1}{s} \] \[ 2Y(s)-sZ(s)+1=\frac{2}{s^2} \]
02

Solve for Z(s)

Solve the first equation for Z(s): \[ Z(s) = \frac{1}{2} \left( \frac{1}{s} - sY(s) \right) \]
03

Substitute Z(s) into the Second Equation

Substitute the expression for Z(s) into the second transformed equation: \[ 2Y(s) - s \left( \frac{1}{2} \left( \frac{1}{s} - sY(s) \right) \right) + 1 = \frac{2}{s^2} \] Simplify this equation: \[ 2Y(s) - \frac{1}{2} + \frac{s^2Y(s)}{2} + 1 = \frac{2}{s^2} \]
04

Combine like terms and solve for Y(s)

Combine like terms to isolate Y(s): \[ 2Y(s) + \frac{s^2}{2}Y(s) + \frac{1}{2}= \frac{2}{s^2} \] \[ Y(s) \left( 2 + \frac{s^2}{2} \right)= \frac{2}{s^2} - \frac{1}{2} \]
05

Simplify Y(s)

Solve for Y(s): \[ Y(s) = \frac{2}{s^2 \left( s^2 + 4 \right)} \]
06

Solve for Z(s)

Plug Y(s) back into the equation for Z(s): \[ Z(s) = \frac{1}{2} \left( \frac{1}{s} - s \frac{2}{s^2 \left( s^2 + 4 \right)} \right) \] Simplify: \[ Z(s) = \frac{1}{2} \left( \frac{1}{s} - \frac{2}{s \left( s^2 + 4 \right)} \right) \]
07

Inverse Laplace Transform

Perform the inverse Laplace Transform to find y(t) and z(t). Using the known transforms: \[ y(t) = L^{-1} \left\{ \frac{2}{s^2 \left( s^2 + 4 \right)} \right\} \] and \[ z(t) = L^{-1} \left\{ \frac{1}{2} \left( \frac{1}{s}- \frac{2}{s \left( s^2 + 4 \right)} \right) \right\} \]
08

Final Solution

Use partial fraction decomposition and standard Laplace inverse pairs. The final solutions are: \[ y(t) = \frac{1}{2}(1 - \cos(2t)) \] and \[ z(t) = \frac{1}{2} \left( 1 + \sin(2t) \right) \]

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Key Concepts

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

Differential Equations
Differential equations are mathematical equations that relate some function with its derivatives. They describe various phenomena such as motion, growth, decay, and oscillations. In this problem, we have two first-order differential equations involving two functions, y(t) and z(t). Each differential equation incorporates derivatives, which add complexity to solving the system. By applying specific techniques such as the Laplace Transform, we can transform these differential equations into simpler algebraic forms that facilitate solving them.
Initial Value Problems
An initial value problem specifies the value of the solution at a particular point, often denoted as the initial condition. In this given problem, initial conditions are provided: \( y(0) = 0 \) and \( z(0) = 1 \). Initial value problems are essential because they enable us to find a unique solution to a differential equation. These conditions are applied after transforming the differential equations using the Laplace Transform. The transformed equations are easier to handle due to the inclusion of initial values.
Inverse Laplace Transform
The Inverse Laplace Transform is a mathematical process that helps revert a function from its Laplace-transformed state back to its original time-domain form. In the final steps of solving the given problem, after obtaining Y(s) and Z(s), we use the inverse transformation to find y(t) and z(t). This process involves using known inverse pairs and sometimes partial fraction decomposition to simplify complex expressions and identify corresponding time-domain functions. The result is the solution to our initial value problem: \ y(t) = \( \frac{1}{2} (1 - \cos(2t)) \) \ and \ z(t) = \( \frac{1}{2} (1 + \sin(2t)) \) \.

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Most popular questions from this chapter

(a) Find the exponential Fourier transform of \(f(x)=e^{-|x|}\) and write the inverse transform. You should find $$ \int_{0}^{\infty} \frac{\cos x x}{x^{2}+1} d x=\frac{\pi}{2} e^{-|x|} $$ (b) Also obtain the result in (a) by using the Fourier cosine transform equations (4.15). (c) Find the Fourier cosine transform of \(f(x)=1 /\left(1+x^{2}\right)\). Hint : Write your result in (b) with \(x\) and \(\alpha\) interchanged.

. \(\quad y^{\prime \prime}+4 y^{\prime}+5 y=\delta\left(t-t_{0}\right)\)

\(f(x)= \begin{cases}1, & \pi / 2<|x|<\pi \\ 0, & \text { otherwise }\end{cases}\)

By using Laplace transforms, solve the following differential equations subject to the given initial conditions. $$ y^{\prime \prime}-4 y=4 e^{2 t}, \quad y_{0}=0, \quad y_{0}^{\prime}=1 $$

(a) Represent as an exponential Fourier transform the function $$ f(x)=\left\\{\begin{array}{cl} \sin x, & 0
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