Short Answer

Find a general solution for the given
linear system using the elimination method of Section 5.2.
$\begin{array}{l} \frac{d^{3} x}{d t^{3}} - x + \frac{d y}{d t} + y = 0 \\ \frac{d x}{d t} - x + y = 0 \end{array}$

_{The general solution is $x (t) = A_{1} + A_{2} e^{t} + A_{3} e^{- t}$}

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Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Elimination method

In the elimination method you either add or subtract the equations to get an equation in one variable. When the coefficients of one variable are opposites you add the equations to eliminate a variable and when the coefficients of one variable are equal you subtract the equations to eliminate a variable.

Step 2: Solving linear system using the elimination method:

We will do the following question on the basis of solving linear system using the elimination method;

$\begin{array}{l} \frac{d^{3} x}{d t^{3}} - x + \frac{d y}{d t} + y = 0 \\ \frac{d x}{d t} - x + y = 0 \\ D^{3} x - x + D y + y = 0 \\ D x - x + y = 0 \\ (D^{3} - 1) x + (D + 1) y = 0 \\ (D - 1) x + y = 0 \\ (D - 1) (D^{3} - 1) x + (D - 1) (D + 1) y - (D^{3} - 1) (D - 1) x + (D^{3} - 1) y = 0 \\ (D^{2} - D^{3}) y = 0 \\ m^{2} - m^{3} = 0 \\ m^{3} (1 - m) = 0 \\ \Rightarrow m = 0, 0, 1 \\ y (t) = C_{1} e^{m_{1} t} + C_{2} e^{m_{2} t} + C_{3} e^{m_{3} t} \\ y (t) = (C_{1} + t C_{2}) + C_{3} e^{t} \\ (D^{3} - 1) x + (D + 1) y - [(D - 1) x + (D + 1) y] = 0 \\ (D^{3} - D) x = 0 \\ m^{3} - m = 0 \\ m (m^{2} - 1) = 0 \\ \Rightarrow m = 0, \pm 1 \\ x (t) = A_{1} e^{m_{1} t} + A_{2} e^{m_{2} t} + A_{3} e^{m_{3} t} \\ = A_{1} + A_{2} e^{t} + A_{3} e^{- t} \end{array}$

Hence, the final answer is:

$x (t) = A_{1} + A_{2} e^{t} + A_{3} e^{- t}$

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Find a general solution for the given
linear system using the elimination method of Section 5.2.
$\begin{array}{l} \frac{d^{3} x}{d t^{3}} - x + \frac{d y}{d t} + y = 0 \\ \frac{d x}{d t} - x + y = 0 \end{array}$

Step by Step Solution

Step 1: Elimination method

Step 2: Solving linear system using the elimination method:

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Fundamentals Of Differential Equations And Boundary Value Problems

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Short Answer

Find a general solution for the givenlinear system using the elimination method of Section 5.2.d3xdt3−x+dydt+y=0dxdt−x+y=0

Step by Step Solution

Step 1: Elimination method

Step 2: Solving linear system using the elimination method:

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Find a general solution for the given
linear system using the elimination method of Section 5.2.
$\begin{array}{l} \frac{d^{3} x}{d t^{3}} - x + \frac{d y}{d t} + y = 0 \\ \frac{d x}{d t} - x + y = 0 \end{array}$