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Use the Laplace transform to solve the given system of differential equations. dy dt dy dt dx dt dx dt + 7x + X + = 1 y = et x(0) = 0, y(0) = 0

Use the Laplace transform to solve the given system of differential equations. dy dt dy dt dx dt dx dt + 7x + X + = 1 y = et x(0) = 0, y(0) = 0

Calculus For The Life Sciences
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter11: Differential Equations
Section11.5: Nonlinear Systems Of Differential Equations
Problem 1YT: YOUR TURN Consider the system of differential equations dx1dt=x1x23x1dx2dt=3x1x26x2 a. Find all...
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### Solving Differential Equations Using Laplace Transforms #### Problem Statement Use the Laplace transform to solve the given system of differential equations: \[ \frac{dx}{dt} + 7x + \frac{dy}{dt} = 1 \] \[ \frac{dx}{dt} - x + \frac{dy}{dt} - y = e^t \] with initial conditions: \[ x(0) = 0, \quad y(0) = 0 \] #### Solution The solutions to the differential equations are: \[ x(t) = \frac{1}{6} - \frac{1}{49} e^t + \frac{1}{7} t e^t - \frac{43}{249} e^{-6t} \] \[ y(t) = -\frac{1}{6} + \frac{13}{49} e^t - \frac{6}{7} e^t - \frac{29}{294} e^{-6t} \] These solutions have been solved with the assumption of initially zero conditions for both functions \(x(t)\) and \(y(t)\). #### Explanation of Solution Steps (Outline): 1. **Transform the Differential Equations**: - Apply the Laplace transform to each differential equation. - Use the properties of Laplace transforms for derivatives. - Incorporate initial conditions during the transformation process. 2. **Solve for Laplace Transforms**: - Rearrange the transformed equations to solve for Laplace transforms of \(X(s)\) and \(Y(s)\). 3. **Inverse Laplace Transform**: - Take the inverse Laplace transform to revert back to the time domain. - Simplify the resulting expressions to obtain \(x(t)\) and \(y(t)\). This systematic approach helps in solving complex differential equations through simplification using the Laplace Transforms.
Transcribed Image Text:### Solving Differential Equations Using Laplace Transforms #### Problem Statement Use the Laplace transform to solve the given system of differential equations: \[ \frac{dx}{dt} + 7x + \frac{dy}{dt} = 1 \] \[ \frac{dx}{dt} - x + \frac{dy}{dt} - y = e^t \] with initial conditions: \[ x(0) = 0, \quad y(0) = 0 \] #### Solution The solutions to the differential equations are: \[ x(t) = \frac{1}{6} - \frac{1}{49} e^t + \frac{1}{7} t e^t - \frac{43}{249} e^{-6t} \] \[ y(t) = -\frac{1}{6} + \frac{13}{49} e^t - \frac{6}{7} e^t - \frac{29}{294} e^{-6t} \] These solutions have been solved with the assumption of initially zero conditions for both functions \(x(t)\) and \(y(t)\). #### Explanation of Solution Steps (Outline): 1. **Transform the Differential Equations**: - Apply the Laplace transform to each differential equation. - Use the properties of Laplace transforms for derivatives. - Incorporate initial conditions during the transformation process. 2. **Solve for Laplace Transforms**: - Rearrange the transformed equations to solve for Laplace transforms of \(X(s)\) and \(Y(s)\). 3. **Inverse Laplace Transform**: - Take the inverse Laplace transform to revert back to the time domain. - Simplify the resulting expressions to obtain \(x(t)\) and \(y(t)\). This systematic approach helps in solving complex differential equations through simplification using the Laplace Transforms.
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