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Section6.1General Solutions of Linear Equations

SubsectionLinear Independence of Functions

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SubsectionThe Wronskian

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SubsectionExercises

1

Verify that the functions \(y_1 = e^{x}\) and \(y_2 = e^{-x}\) are solutions to the DE: \begin{equation*} y'' - y = 0 \end{equation*} then find a particular solution of the form \(y = c_1 y_1 + c_2 y_2\) which satisfies the initial conditions \(y(0) = 0, y'(0) = 5\text{.}\)

Answer
2

Verify that the functions \(y_1 = e^{5x}\) and \(y_2 = xe^{5x}\) are solutions to the DE: \begin{equation*} y'' - 10 y' + 25 y = 0 \end{equation*} then find a particular solution of the form \(y = c_1 y_1 + c_2 y_2\) which satisfies the initial conditions \(y(0) = 3, y'(0) = 13\text{.}\)

Answer
3

Verify that the functions \(y_1 = e^{-3x}\cos 2x\) and \(y_2 = e^{-3x} \sin 2x\) are solutions to the DE: \begin{equation*} y'' + 6 y' + 13 y = 0 \end{equation*} then find a particular solution of the form \(y = c_1 y_1 + c_2 y_2\) which satisfies the initial conditions \(y(0) = 2, y'(0) = 0\text{.}\)

Answer
4

Let \(x_p\) be a solution to the nonhomogeneous matrix equation: \begin{equation} A \vec{x} = \vec{b}, \label{eqn-matrix-nonhomogeneous}\tag{6.1.1} \end{equation} and let \(x_h\) be a solution to the the associated homogeneous equation: \begin{equation} A \vec{x} = \vec{0}, \label{eqn-matrix-homogeneous}\tag{6.1.2} \end{equation} Show that \(\vec{x} = \vec{x}_h + \vec{x}_p\) is also a solution the the nonhomogeneous equation (6.1.1).

Hint
5

Let \(y_p\) be a solution to the nonhomogeneous equation: \begin{equation} y'' + py' + qy = f(x), \label{eqn-second-order-nonhomogeneous}\tag{6.1.3} \end{equation} and let \(y_h\) be a solution to the the associated homogeneous equation: \begin{equation} y'' + py' + qy = 0. \label{eqn-second-order-homogeneous}\tag{6.1.4} \end{equation} Show that \(y = y_h + y_p\) is also a solution the the nonhomogeneous equation (6.1.3).

Hint
6

Show directly that the given functions are linearly dependent on the real line. Find a nontrivial linear combination of the given function that equals zero for all values of \(x\text{.}\) \begin{equation*} f(x) = 5 \quad g(x) = 2-3x^2 \quad h(x) = 10 + 15x^2 \end{equation*}

7

Use the Wronskian to prove that the following functions are linearly independent on the interval \((-\infty, \infty)\text{.}\) \begin{equation*} f(x) = 1 \quad g(x) = x \quad h(x) = x^2 \end{equation*}

8

Use the Wronskian to prove that the following functions are linearly independent on the interval \((-\infty, \infty)\text{.}\) \begin{equation*} f(x) = e^x \quad g(x) = e^{2x} \quad h(x) = e^{3x} \end{equation*}