Question

Proceed as in Problem 7 to transform the given system into a single equation of second order. Then find \(x_{1}\) and \(x_{2}\) that also satisfy the given initial conditions. Finally, sketch the graph of the solution in the \(x_{1} x_{2}\) -plane for \(t \geq 0 .\) \(\begin{array}{ll}{x_{1}^{\prime}=3 x_{1}-2 x_{2},} & {x_{1}(0)=3} \\\ {x_{2}^{\prime}=2 x_{1}-2 x_{2},} & {x_{2}(0)=\frac{1}{2}}\end{array}\)

Short answer

Answer: x2(t) = x1(t) * ((1/2 + t) / 3)

Step-by-step solution

Transform the given system into a single second-order equation

To transform the given system into a single second-order equation, we differentiate one of the equations with respect to \(t\) and substitute the other equation to eliminate one of the variables. Let's differentiate the first equation: \(x_1^{\prime\prime} = 3x_1^{\prime} - 2x_2^{\prime}\) Now, substitute the values of \(x_1^{\prime}\) and \(x_2^{\prime}\) from the given equations: \(x_1^{\prime\prime} = 3(3x_1 - 2x_2) - 2(2x_1 - 2x_2)\) Simplify the equation: \(x_1^{\prime\prime} = 7x_1 - 6x_2\)

Solve the second-order equation

To solve the second-order equation \(x_1^{\prime\prime} = 7x_1 - 6x_2\), we will consider \(x_1^{\prime} = x_3\). This way, we can rewrite the equations as follows: \(x_1^{\prime} = x_3\) \(x_3^{\prime} = 7x_1 - 6x_2\) Now, solve for \(x_1\) and \(x_2\) using the initial conditions: \(x_1(0) = 3\) \(x_1^{\prime}(0) = x_3(0) = 3x_1(0) - 2x_2(0) = 3 \cdot 3 - 2 \cdot \frac{1}{2} = 8\)

Find \(x_1\) and \(x_2\) satisfying the initial conditions

With the initial conditions known, we can integrate the second-order equation to find \(x_1\) and \(x_2\): \(x_1(t) = 3e^{2t}\) \(x_2(t) = \frac{1}{2}e^{2t} + te^{2t}\)

Sketch the graph of the solution in the \(x_1 x_2\)-plane

To sketch the graph of the solution \(x_1(t) = 3e^{2t}\) and \(x_2(t) = \frac{1}{2}e^{2t} + te^{2t}\) in the \(x_1 x_2\)-plane, we can eliminate \(t\) by taking \(x_2\) as a function of \(x_1\). Divide the second equation by the first equation: \(\frac{x_2(t)}{x_1(t)} = \frac{\frac{1}{2}e^{2t} + te^{2t}}{3e^{2t}} = \frac{\frac{1}{2} + t}{3}\) Now, solving for \(x_2(t)\) as a function of \(x_1(t)\): \(x_2(t) = x_1(t) \cdot \left(\frac{\frac{1}{2} + t}{3}\right)\) To sketch the graph, plot the function \(x_2(t)\) as a function of \(x_1(t)\). The graph should show an exponentially increasing curve that starts at the initial conditions \((3, \frac{1}{2})\) and has an increasing slope as the time \(t\) increases.