Linear Second-Order Equations

Verify that for $\mathit{b}\mathbf{=}\mathbf{0}$ and ${\mathit{F}}_{\mathit{e}\mathit{x}\mathit{t}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$, equation (3) has a solution of the form $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{cos\omega t}\mathbf{,}\mathbf{\omega }\mathbf{=}\sqrt{\mathbf{k}\mathbf{/}\mathbf{m}}$

If ${\mathit{F}}_{\mathit{e}\mathit{x}\mathit{t}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$, equation (3) becomes $\mathit{m}\mathit{y}\text{'}\text{'}\mathbf{+}\mathit{b}\mathit{y}\text{'}\mathbf{+}\mathit{k}\mathit{y}\mathbf{=}\mathbf{0}$. For this equation, verify the following:

(a) If $\mathit{y}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ is a solution so is $\mathit{c}\mathit{y}\mathbf{\left(}\mathit{t}\mathbf{\right)}$, for any constant c.

(b) If ${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ and ${\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$ are solutions, so is their sum ${\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{+}{\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}$.

Question: Show that if ${\mathbf{F}}_{\mathbf{ext}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{k}\mathbf{=}\mathbf{9}$, and , then the equation   has the "critically damped" solutions ${\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}$ and ${\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{t}{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}$ . What is the limit of these solutions as $\mathbf{t}\mathbf{\to }\mathbf{\infty }$?

Question: Find a general solution to the given differential equation.

$\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{7}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{0}$

Verify that $\mathit{y}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{3}\mathit{t}\mathbf{+}\mathbf{2}\mathit{c}\mathit{o}\mathit{s}\mathbf{3}\mathit{t}$ is a solution to the initial value problem

$\mathbf{2}\mathit{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{18}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{;}\mathbf{ }\mathbf{ }\mathbf{ }\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathbf{ }\mathbf{ }\mathbf{ }\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{3}$

Find the maximum of $\left|y\left(t\right)\right|$ for $\mathbf{-}\mathbf{\infty }\mathbf{<}\mathit{t}\mathbf{<}\mathbf{\infty }$.

Question: Verify that the exponentially damped sinusoid $\mathbf{y}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{-}\mathbf{3}\mathbf{t}}\mathbf{sin}\mathbf{(}\sqrt{\mathbf{3}\mathbf{t}}\mathbf{)}$is a solution to the equation  if ${\mathbf{F}}_{\mathbf{ext}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{b}\mathbf{=}\mathbf{6}\mathbf{ }$and k= 12. What is the limit of this solution as $\mathbf{ }\mathbf{\infty }$ ?

Question: An external force $\mathbf{F}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{cos}\mathbf{2}\mathbf{t}$ is applied to a mass-spring system with,$\mathbf{m}\mathbf{=}\mathbf{1}\mathbf{,}\mathbf{b}\mathbf{=}\mathbf{0}$ and, $\mathbf{k}\mathbf{=}\mathbf{4}$which is initially at rest; i.e., $\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}$. Verify that $\mathbf{y}\left(t\right)\mathbf{=}\frac{\mathbf{1}}{\mathbf{2}}\mathbf{tsin}\mathbf{2}\mathbf{t}\mathbf{ }\mathbf{,}\mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}\mathbf{,}\mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{0}$gives the motion of this spring. What will eventually (asincreases) happen to the spring?

Question: Find a synchronous solution of the form $\mathbf{Acos\Omega t}\mathbf{+}\mathbf{Bsin\Omega t}$ to the given forced oscillator equation using the method of Example   to solve for  A and B $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{5}\mathbf{sin}\mathbf{3}\mathbf{t}\mathbf{,}\mathbf{\Omega }\mathbf{=}\mathbf{3}$.

 .

Question: Find a synchronous solution of the form $\mathbf{Acos\Omega t}\mathbf{+}\mathbf{Bsin\Omega t}$to the given forced oscillator equation using the method of Example  to solve for  A and B $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{5}\mathbf{y}\mathbf{=}\mathbf{-}\mathbf{50}\mathbf{sin}\mathbf{5}\mathbf{t}\mathbf{,}\mathbf{\Omega }\mathbf{=}\mathbf{5}$ .

 .

Question: Find a synchronous solution of the form $\mathbf{Acos\Omega t}\mathbf{+}\mathbf{Bsin\Omega t}$ to the given forced oscillator equation using the method of Example 4  to solve for A  and B$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{4}\mathbf{y}\mathbf{=}\mathbf{6}\mathbf{cos}\mathbf{2}\mathbf{t}\mathbf{+}\mathbf{8}\mathbf{sin}\mathbf{2}\mathbf{t}\mathbf{,}\mathbf{\Omega }\mathbf{=}\mathbf{2}$  .

 .

Undamped oscillators that are driven at resonance have unusual (and nonphysical) solutions.

1. To investigate this, find the synchronous solution $\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathit{\Omega }\mathit{t}\mathbf{+}\mathit{B}\mathit{s}\mathit{i}\mathit{n}\mathit{\Omega }\mathit{t}$ to the generic forced oscillator equation $\mathbf{\left(}\mathbf{7}\mathbf{\right)}\mathit{m}\mathit{y}\text{'}\text{'}\mathbf{+}\mathit{b}\mathit{y}\text{'}\mathbf{+}\mathit{k}\mathit{y}\mathbf{=}\mathit{c}\mathit{o}\mathit{s}\mathit{\Omega }\mathit{t}$.
2. Sketch graphs of the coefficients A and B, as functions of $\mathit{\Omega }$ for $\mathit{m}\mathbf{=}\mathbf{1}\mathbf{,}\mathit{b}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{,}\mathit{k}\mathbf{=}\mathbf{25}$.
3. Now set $\mathit{b}\mathbf{=}\mathbf{0}$ in your formulas for A and B and re-sketch the graphs in part (b), with $\mathit{m}\mathbf{=}\mathbf{1}$, and $\mathit{k}\mathbf{=}\mathbf{25}$. What happens at $\mathit{\Omega }\mathbf{=}\mathbf{5}$? Notice that the amplitudes of the synchronous solutions grow without bound as $\mathit{\Omega }$ approaches 5.
4. Show directly, by substituting the form $\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathit{\Omega }\mathit{t}\mathbf{+}\mathit{B}\mathit{s}\mathit{i}\mathit{n}\mathit{\Omega }\mathit{t}$ into equation (7), that when $\mathit{b}\mathbf{=}\mathbf{0}$ there are no synchronous solutions if $\mathit{\Omega }\mathbf{=}\sqrt{\mathit{k}\mathbf{/}\mathit{m}}$.
5. Verify that ${\mathbf{\left(}\mathbf{2}\mathit{m}\mathit{\Omega }\mathbf{\right)}}^{\mathbf{-}\mathbf{1}}\mathit{t}\mathit{s}\mathit{i}\mathit{n}\mathit{\Omega }\mathit{t}$ solves equation (7) when $\mathit{b}\mathbf{=}\mathbf{0}$ and $\mathit{\Omega }\mathbf{=}\sqrt{\mathit{k}\mathbf{/}\mathit{m}}$. 

Notice that this nonsynchronous solution grows in time, without bound.

Clearly one cannot neglect damping in analyzing an oscillator forced at resonance, because otherwise the solutions, as shown in part (e), are nonphysical. This behavior will be studied later in this chapter.

Question: Find a geneQuestion: Find a general solution to the given differential equation.ral solution to the given differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: Find a general solution to the given differential equation.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{5}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{6}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: Find a general solution to the given differential equation.

$\mathbf{6}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{2}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: find a general solution to the given differential equation.

$\mathbf{z}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{z}\mathbf{\text{'}}\mathbf{-}\mathbf{z}\mathbf{=}\mathbf{0}$

Question: find a general solution to the given differential equation.

$\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{4}\mathbf{y}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: find a general solution to the given differential equation $\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{-}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{11}\mathbf{y}\mathbf{=}\mathbf{0}$.

Question: find a general solution to the given differential equation.

$\mathbf{4}\mathbf{w}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{20}\mathbf{w}\mathbf{\text{'}}\mathbf{+}\mathbf{25}\mathbf{w}\mathbf{=}\mathbf{0}$

Question: Find a general solution to the given differential equation.

$\mathbf{3}\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{11}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{7}\mathbf{y}\mathbf{=}\mathbf{0}$

Question: Solve the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{2}\mathbf{y}\mathbf{\text{'}}\mathbf{-}\mathbf{8}\mathbf{y}\mathbf{=}\mathbf{0}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{3}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{-}\mathbf{12}$

Question: Solve the given initial value problem.

$\mathbf{y}\mathbf{\text{'}}\mathbf{\text{'}}\mathbf{+}\mathbf{y}\mathbf{\text{'}}\mathbf{=}\mathbf{0}\mathbf{;} \mathbf{y}\left(\mathbf{0}\right)\mathbf{=}\mathbf{2}\mathbf{,} \mathbf{y}\mathbf{\text{'}}\left(\mathbf{0}\right)\mathbf{=}\mathbf{1}$

In Problems 13–20, solve the given initial value problem.

${\mathit{y}}^{\mathbf{"}}\mathbf{-}\mathbf{4}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{+}\mathbf{3}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{:}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{1}\mathbf{,}{\mathit{y}}^{\mathbf{\text{'}}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{1}}{\mathbf{3}}$

In Problems 13–20, solve the given initial value problem.

y"-4y'-5y = 0 : y(-1) = 3,y'(-1) = 9

In Problems 13–20, solve the given initial value problem.

$\mathit{y}\mathbf{"}\mathbf{-}\mathbf{6}\mathit{y}\mathbf{\text{'}}\mathbf{+}\mathbf{9}\mathit{y}\mathbf{=}\mathbf{0}\mathbf{:}\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{,}\mathit{y}\mathbf{\text{'}}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\frac{\mathbf{25}}{\mathbf{3}}$

In Problems 13–20, solve the given initial value problem.

z" - 2z' -2z = 0 : z(0) = 0, z'(0) = 3

In Problems 13–20, solve the given initial value problem.

y" + 2y' + y = 0 : y(0) = 1, y'(0) = -3

In Problems 13–20, solve the given initial value problem.

y" - 4y' + 4y = 0 : y(1) = 1, y'(1) =1

First-Order Constant-Coefficient Equations.

In Problems 22–25, use the method described in Problem 21to find a general solution to the given equation.

22. 3Y' - 7Y = 0

In Problems 22–25, use the method described in Problem 21 to find a general solution to the given equation.

23. 5Y' + 4Y = 0

In Problems 22–25, use the method described in Problem 21 to find a general solution to the given equation.

24. 3z' + 11z = 0

In Problems 22–25, use the method described in Problem 21 to find a general solution to the given equation.

25. 6w' - 13w = 0

Boundary Value Problems. When the values of a solution to a differential equation are specified at two different points, these conditions are called boundary conditions. (In contrast, initial conditions specify the values of a function and its derivative at the same point.) The purpose of this exercise is to show that for boundary value problems there is no existence–uniqueness theorem that is analogous to Theorem 1. Given that every solution to (17)y" + y = 0 is of the form y(t) = c₁cost + c₂sint, where c₁ and c₂ are arbitrary constants, show that

(a) There is a unique solution to (17) that satisfies the boundary conditionsand $\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{y}\left(\frac{\pi }{2}\right)\mathbf{=}\mathbf{0}$.

(b) There is no solution to (17) that satisfies and $\mathit{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{ }\mathit{a}\mathit{n}\mathit{d}\mathbf{ }\mathit{y}\mathbf{\left(}\mathbf{\pi }\mathbf{\right)}\mathbf{=}\mathbf{0}$.

(c) There are infinitely many solutions to (17) that satisfy y()) = 2 and $\mathit{y}\mathbf{\left(}\mathbf{\pi }\mathbf{\right)}\mathbf{=}\mathbf{-}\mathbf{2}$.

In Problems 27–32, use Definition 1 to determine whether the functions y₁ and y₂ are linearly dependent on the interval (0, 1).

27. y₁(t) = costsint, y₂(t) = sin2t

In Problems 27–32, use Definition 1 to determine whether the functions y₁ and y₂ are linearly dependent on the interval (0, 1).

28. y₁(t) = e^3t, y₂(t) = e^-4t

In Problems 27–32, use Definition 1 to determine whether the functions y₁ and y₂ are linearly dependent on the interval (0, 1).

29. y₁(t) = te^2t, y₂(t) = e^2t

In Problems 27–32, use Definition 1 to determine whether the functions y₁ and y₂ are linearly dependent on the interval (0, 1).

30. y₁(t) = t²cos(lnt), y₂(t) = t²sin(lnt)

In Problems 27–32, use Definition 1 to determine whether the functions y₁ and y₂ are linearly dependent on the interval (0, 1).

31. y₁(t) = tan²t - sec²t,y₂(t) = 3

In Problems 27–32, use Definition 1 to determine whether the functions y₁ and y₂ are linearly dependent on the interval (0, 1).

32. y₁(t)= 0, y₂(t) = e^t

Explain why two functions are linearly dependent on an interval I if and only if there exist constants c₁ and c₂, not both zero, such that for ${\mathit{c}}_{\mathbf{1}}{\mathit{y}}_{\mathbf{1}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{y}}_{\mathbf{2}}\mathbf{\left(}\mathit{t}\mathbf{\right)}\mathbf{=}\mathbf{0}$ all t in I.

Wronskian. For any two differentiable functions y₁ and y₂, the function (18) W[y₁,y₂](t) = y₁(t)_,y'₂(t) - y'₁(t),y₂(t) is called the Wronskian of y₁ and y₂. This function plays a crucial role in the proof of Theorem 2.

Linear Dependence of Three Functions.

Three functions y₁(t), y₂(t) and y₃(t) are said to be linearly dependent on an interval if, on l, at least one of these functions is a linear combination of the remaining two e.g., if y₁(t) = c₁y₂(t) + c₂y₃(t). Equivalently (compare Problem), y₁,y₂ and y₃ are linearly dependent on l if there exist constants C₁,C₂ and C₃, not all zero, such that C₁y₁(t) + C₂y₂(t) + C₃y₃(t) = 0 for all t in l. Otherwise, we say that these functions are linearly independent on. For each of the following, determine whether the given three functions are linearly dependent or linearly independent on $\left(-\infty ,\infty \right)$:

$\begin{array}{rcl}\mathbf{\left(}\mathbf{a}\mathbf{\right)}\mathbf{ }{\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}& \mathbf{=}& \mathbf{1}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{t}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{3}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}\mathbf{.}\\ \mathbf{\left(}\mathbf{b}\mathbf{\right)}\mathbf{ }{\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}& \mathbf{=}& \mathbf{-}\mathbf{3}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{5}{\mathbf{sin}}^{\mathbf{2}}\mathbf{t}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{3}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{cos}}^{\mathbf{2}}\mathbf{t}\mathbf{.}\\ \mathbf{\left(}\mathbf{c}\mathbf{\right)}\mathbf{ }{\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}& \mathbf{=}& {\mathbf{e}}^{\mathbf{t}}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{te}}^{\mathbf{t}}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{3}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{t}}^{\mathbf{2}}{\mathbf{e}}^{\mathbf{t}}\mathbf{.}\\ \mathbf{\left(}\mathbf{d}\mathbf{\right)}\mathbf{ }{\mathbf{y}}_{\mathbf{1}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}& \mathbf{=}& {\mathbf{e}}^{\mathbf{t}}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{2}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}{\mathbf{e}}^{\mathbf{-}\mathbf{t}}\mathbf{,}\mathbf{ }\mathbf{ }{\mathbf{y}}_{\mathbf{3}}\mathbf{\left(}\mathbf{t}\mathbf{\right)}\mathbf{=}\mathbf{cosht}\mathbf{.}\end{array}$

Using the definition in Problem 35, prove that if r₁ , r₂ and r₃ are distinct real numbers, then the functions ${\mathit{e}}^{{\mathit{r}}_{\mathbf{1}}\mathit{t}}\mathbf{,}{\mathit{e}}^{{\mathit{r}}_{\mathbf{2}}\mathit{t}}$, and ${\mathit{e}}^{{\mathit{r}}_{\mathbf{3}}\mathit{t}}$ are linearly independent on $\mathbf{(}\mathbf{-}\infty \mathbf{,}\infty \mathbf{)}$.

[Hint: Assume to the contrary that, say, ${\mathit{e}}^{{\mathit{r}}_{\mathbf{1}}\mathit{t}}\mathbf{=}{\mathit{c}}_{\mathbf{1}}{\mathit{e}}^{{\mathit{r}}_{\mathbf{2}}\mathit{t}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{{\mathit{r}}_{\mathbf{3}}\mathit{t}}$ for all t. Divide by ${\mathit{e}}^{{\mathit{r}}_{\mathbf{2}}\mathit{t}}$ to get ${\mathit{e}}^{\left({\mathit{r}}_{\mathbf{1}}\mathbf{-}{\mathit{r}}_{\mathbf{2}}\right)\mathit{t}}\mathbf{=}{\mathit{c}}_{\mathbf{1}}\mathbf{+}{\mathit{c}}_{\mathbf{2}}{\mathit{e}}^{\left({\mathit{r}}_{\mathbf{3}}\mathbf{-}{\mathit{r}}_{\mathbf{2}}\right)\mathit{t}}$ and then differentiate to deduce that ${\mathit{e}}^{\left({\mathit{r}}_{\mathbf{1}}\mathbf{-}{\mathit{r}}_{\mathbf{2}}\right)\mathit{t}}$ and ${\mathit{e}}^{\left({\mathit{r}}_{\mathbf{3}}\mathbf{-}{\mathit{r}}_{\mathbf{2}}\right)\mathit{t}}$ are linearly dependent, which is a contradiction. (Why?)]

Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.

$\mathit{y}\text{'}\text{'}\text{'}\mathbf{+}\mathit{y}\text{'}\text{'}\mathbf{-}\mathbf{6}\mathit{y}\text{'}\mathbf{+}\mathbf{4}\mathit{y}\mathbf{=}\mathbf{0}$.

Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it. $\mathit{y}\text{'}\text{'}\text{'}\mathbf{-}\mathbf{6}\mathit{y}\text{'}\text{'}\mathbf{-}\mathit{y}\text{'}\mathbf{+}\mathbf{6}\mathit{y}\mathbf{=}\mathbf{0}$.

Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.  $\mathit{z}\text{'}\text{'}\text{'}\mathbf{+}\mathbf{2}\mathit{z}\text{'}\text{'}\mathbf{-}\mathbf{4}\mathit{z}\text{'}\mathbf{-}\mathbf{8}\mathit{z}\mathbf{=}\mathbf{0}$

Find three linearly independent solutions (see Problem 35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it. $\mathit{y}\text{'}\text{'}\text{'}\mathbf{-}\mathbf{7}\mathit{y}\text{'}\text{'}\mathbf{+}\mathbf{7}\mathit{y}\text{'}\mathbf{+}\mathbf{15}\mathit{y}\mathbf{=}\mathbf{0}$.

Find three linearly independent solutions (see Problem35) of the given third-order differential equation and write a general solution as an arbitrary linear combination of it.

y"' + 3y" - 4y' - 12y = 0.

(True or False) If f₁(t), f₂(t), f₃(t) are three functions defined on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$ that are pairwise linearly independent on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$, then f₁(t), f₂(t), f₃(t) form a linearly independent set on $\mathbf{(}\mathbf{-}\mathbf{\infty }\mathbf{,}\mathbf{\infty }\mathbf{)}$. Justify your answer.

Solve the initial value problem:

y"' - y' = 0;   y(0) = 2,

y'(0) = 3,   y"(0) = -1