Higher-order homogeneous equations

Section 2.1 Higher-order homogeneous equations

Remark 2.1.1.

Suppose an object with mass \(m\) (in kilograms \(kg\)) hangs vertically at equilibrium on a spring. When the spring is stretched \(y>\) meters, there is a corresponding restoring force \(F_R\) (in newtons \(N\)). When the object moves at a velocity \(v\text{,}\) surrounding medium exerts a corresponding damping force \(F_D\) (in newtons \(N\)). Initially, the object is released with an initial poisition \(y_0\) and an initial velocity \(y_1\text{.}\) Write a differential equation for the spring mass when there is an external force by \(f(t)\text{.}\)

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Here, we treat the downward direction as the positive direction.

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Definition 2.1.2. Higher-order linear ODEs.

An \(n\)th-order linear ODE is of the form

\begin{equation} a_n(t)y^{(n)}+a_{n-1}(t)y^{(n-1)}+\ldots+a_2(t)y''+a_1(t)y'+a_0(t)y=g(t),\tag{2.1.1} \end{equation}

where \(g\) is called a forcing function, input, or control. If \(g(t)=0\) for all \(t\) (on some interval \(I\)), then we call (2.1.1) homogeneous:

\begin{equation} a_n(t)y^{(n)}+a_{n-1}(t)y^{(n-1)}+\ldots+a_2(t)y''+a_1(t)y'+a_0(t)y=0.\tag{2.1.2} \end{equation}

If \(g(t)\neq 0\) for even one value \(t\text{,}\) then (2.1.1) is nonhomogeneous.

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Theorem 2.1.3. Existence and uniqueness for second-order linear IVPs.

Consider the IVP

\begin{equation} a_2(t)y''+a_1(t)y'+a_0y=g(t), \quad y(t_0)=y_0, y'(t_0)=y_1.\tag{2.1.3} \end{equation}

Rewriting in standard form as

\begin{equation*} y''+P(t)y'+Q(t)y=f(t), \end{equation*}

(2.1.3) is guaranteed to have a unique solution if \(P\text{,}\) \(Q\text{,}\) and \(f\) are all continuous on an open interval \(I\) containing \(t_0\text{.}\) The largest such interval \(I\) is called the interval of definition for the IVP.

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Example 2.1.4.

Find the largest interval where the IVP is guaranteed a unique solution.

\(\displaystyle (t^2-4)y''+ty=\sqrt{t}, \quad y(3)=4, y'(3)=0\)

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\(\displaystyle t^2y''+3y'+(1-2t)y=\arcsin(t), \quad y(1/2)=1, y'(1/2)=4\)

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Definition 2.1.5. Homogeneous solutions.

An explicit solution to (2.1.2) on some interval \(I\) is a function that, when substitued into (2.1.2), results in an identity.

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Example 2.1.6.

Consider the differential equation \(ty''-(t+1)y'+y=0\text{.}\) Determine the following are solutions for \(t>0\text{:}\)

\(\displaystyle y(t)=t\)

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\(\displaystyle y(t)=e^t\)

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Example 2.1.7.

Consider the differential equation \(y'''-2y''-y'+2y=0\text{.}\) Given that \(y_1(t)=e^t\) and \(y_2(t)=e^{-t}\) are solutions on \((-\infty,\infty)\text{,}\) show that the following are also solutions on \((-\infty,\infty)\text{:}\)

\(\displaystyle y(t)=0\)

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\(y(t)=c_1 y_1(t)\) for an arbitrary constant \(c_1\)

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\(y(t)=c_1y_1(t)+c_2y_2(t)\) for arbitrary constants \(c_1, c_2\)

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Remark 2.1.8. Hyperbolic trigonometric functions.

Recall the hyperbolic sine is \(\sinh(t):=\dfrac{e^t-e^{-t}}{2}\) and the hyperbolic cosine is \(\cosh(t)=\dfrac{e^t+e^{-t}}{2}\text{.}\)

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Theorem 2.1.9. Superposition principle.

Let \(y_1, y_2, \ldots, y_k\) be solutions to (2.1.2) on an interval \(I\text{.}\) Then the linear combination

\begin{equation*} y(t)=c_1y_1(t)+c_2y_2(t)+\ldots+c_ky_k(t), \end{equation*}

for arbitrary constants \(c_1,\ldots,c_k\) is also a solution to (2.1.2) on \(I\text{.}\)

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Remark 2.1.10.

Recall that for linear equations, a family of solutions represents all possible solutions on \(I\text{.}\) Given a linear homogeneous ODE and a set of functions contiuous on \(I\text{,}\) we have the following questions:

Are they really solutions?

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Are they "different" solutions?

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Do we have all solutions?

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Definition 2.1.11. Linear independence.

A set of functions \(f_1,f_2,\ldots,f_k\) are called linearly independent on an interval \(I\) if and only if the equation

\begin{equation*} c_1f_1(t)+c_2f_2(t)+\ldots+c_k f_k(t)=0 \end{equation*}

holds for all \(t\) in \(I\) only when \(c_1=c_2=\ldots=c_k=0\text{.}\) If there are nonzero constants \(c_1,\ldots,c_k\) such that for all \(t\) in \(I\text{,}\) \(c_1f_1(t)+\ldots+c_kf_k(t)=0\text{,}\) then we say that \(f_1,\ldots,f_k\) are linearly dependent.

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Example 2.1.12.

Let \(f(t)=1\text{,}\) \(g(t)=t\text{,}\) and \(h(t)=2t+3\text{.}\)

Show that \(f\) and \(g\) are linearly independent on \((-\infty,\infty)\text{.}\)

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Show that \(f\) and \(h\) are linearly independent on \((-\infty,\infty)\text{.}\)

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Show that \(g\) and \(h\) are linearly independent on \((-\infty,\infty)\text{.}\)

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Determine if \(f\text{,}\) \(g\text{,}\) and \(h\) are linearly independent on \((-\infty,\infty)\text{.}\)

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Example 2.1.13.

Let \(f(t)=t\) and \(g(t)=|t|\text{.}\)

Determine if \(f\) and \(g\) are linearly independent on \((0,\infty)\text{.}\)

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Determine if \(f\) and \(g\) are linearly independent on \((-\infty,0)\text{.}\)

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Determine if \(f\) and \(g\) are linearly independent on any interval containing \(t=0\text{.}\)

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Theorem 2.1.14. The Wronskian.

The set of functions \(f_1,\ldots,f_n\) in \(C^{n-1}(I)\) are linearly independent on \(I\) if and only if the Wronskian

\begin{equation*} W\{f_1,\ldots,f_n\}(t):=\mathrm{det} \begin{bmatrix} f_1(t) & f_2(t) & \ldots & f_n(t) \\ f_1'(t) & f_2'(t) & \ldots f_n'(t) \\ \vdots & \vdots & \ddots & \vdots \\ f_1^{(n-1)}(t) & f_2^{(n-1)}(t) & \ldots & f_n^{(n-1)}(t) \end{bmatrix} \neq 0. \end{equation*}

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Example 2.1.15.

Determine whether \(\{t,\ln(t)\}\) is a linearly independent set on \((0,\infty)\text{.}\)

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Definition 2.1.16. Fundamental set of solutions (FSS).

Let \(y_1,y_2,\ldots,y_n\) be functions in \(C^n(I)\text{.}\) Then \(y_1,y_2,\ldots,y_n\) form a fundamental set of solutions to (2.1.2) on \(I\) if

\(y_1,y_2,\ldots,y_n\) are solutions to (2.1.2) on \(I\text{,}\)

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\(y_1,y_2,\ldots,y_n\) are linearly independent on \(I\text{,}\) and

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the number of linearly independent solutions on \(I\) are equal to the order of (2.1.2).

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Definition 2.1.17. General solution.

Suppose \(y_1,y_2,\ldots,y_n\) form a fundamental set of solutions to (2.1.2) on \(I\text{.}\) Then the linear combination

\begin{equation*} y(t)=c_1y_1(t)+c_2y_2(t)+\ldots+c_n y_n(t), \end{equation*}

where \(c_1,c_2,\ldots,c_n\) are arbitrary constants is called a general solution to (2.1.2) on \(I\text{.}\)

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Example 2.1.18.

Consider the differential equation \(ty''-(t+1)y'+y=0\) on \((0,\infty)\text{.}\)

Do \(f(t)=e^t\) and \(g(t)=t\) form a fundamental set of solutions to the differential equation on \((0,\infty)\text{?}\) State why or why not. If so, then write a general solution.

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Do \(f(t)=e^t\) and \(g(t)=4e^t\) form a fundamental set of solutions to the differential equation on \((0,\infty)\text{?}\) State why or why not. If so, then write a general solution.

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Do \(f(t)=e^t\) and \(g(t)=t+1\) form a fundamental set of solutions to the differential equation on \((0,\infty)\text{?}\) State why or why not. If so, then write a general solution.

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Example 2.1.19.

Consider the following differential equations. Along with each equation is a given fact. Give a reason why we may or we may not have a fundamental set of solutions.

\((t^2-9)y'''-4\sin(t)y'-y=0, (3,\infty)\)
Known: Three solutions on \((3,\infty)\)
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What can go wrong?
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\(t^2y''+(1-2t)y'+8y=0, (0,\infty)\)
Known: Two linearly independent solutions on \((0,\infty)\)
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What can go wrong?
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\(y^{(4)}+ty'+(1-4t)y=0, (-\infty,\infty)\)
Known: Three linearly independent solutions on \((-\infty,\infty)\)
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What can go wrong?
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\((t-2)y''+(2t-3)y'+4y=0, (2,\infty)\)
Known: Three linearly independent solutions on \((2,\infty)\)
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What can go wrong?
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