Advanced Engineering Mathematics

Dennis G. Zill, Warren S. Wright

Chapter 3

Higher-Order Differential Equations - all with Video Answers

Educators

Section 1

Theory of Linear Equations

Problem 1

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
$$
y=c_{1} e^{x}+c_{2} e^{-x},(-\infty, \infty) ; y^{\prime \prime}-y=0, y(0)=0, y^{\prime}(0)=1
$$

Hast Aggarwal

Hast Aggarwal

Numerade Educator

Problem 2

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
$$
\begin{aligned}
&y=c_{1} e^{4 x}+c_{2} e^{-x},(-\infty, \infty) ; y^{\prime \prime}-3 y^{\prime}-4 y=0, y(0)=1, \\
&y^{\prime}(0)=2
\end{aligned}
$$

Hast Aggarwal

Numerade Educator

Problem 3

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
$$
\begin{aligned}
&y=c_{1} x+c_{2} x \ln x,(0, \infty) ; x^{2} y^{\prime \prime}-x y^{\prime}+y=0, y(1)=3 \\
&y^{\prime}(1)=-1
\end{aligned}
$$

Hast Aggarwal

Numerade Educator

Problem 4

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
$$
\begin{aligned}
&y=c_{1}+c_{2} \cos x+c_{3} \sin x,(-\infty, \infty) ; y^{\prime \prime \prime}+y^{\prime}=0 \\
&y(\pi)=0, y^{\prime}(\pi)=2, y^{\prime \prime}(\pi)=-1
\end{aligned}
$$

Hast Aggarwal

Numerade Educator

Problem 5

Given that $y=c_{1}+c_{2} x^{2}$ is a two-parameter family of solutions of $x y^{\prime \prime}-y^{\prime}=0$ on the interval $(-\infty, \infty)$, show that constants $c_{1}$ and $c_{2}$ cannot be found so that a member of the family satisfies the initial conditions $y(0)=0, y^{\prime}(0)=1$. Explain why this does not violate Theorem 3.1.1.

Hast Aggarwal

Numerade Educator

Problem 6

Find two members of the family of solutions in Problem 5 that satisfy the initial conditions $y(0)=0, y^{\prime}(0)=0$.

Mahnoor Khan

Numerade Educator

Problem 7

Given that $x(t)=c_{1} \cos \omega t+c_{2} \sin \omega t$ is the general solution
of $x^{\prime \prime}+\omega^{2} x=0$ on the interval $(-\infty, \infty)$, show that a solution satisfying the initial conditions $x(0)=x_{0}, x^{\prime}(0)=x_{1}$, is given by
$$
x(t)=x_{0} \cos \omega t+\frac{x_{1}}{\omega} \sin \omega t .
$$

Mahnoor Khan

Numerade Educator

Problem 8

Use the general solution of $x^{\prime \prime}+\omega^{2} x=0$ given in Problem 7 to show that a solution satisfying the initial conditions $x\left(t_{0}\right)=$ $x_{0,} x^{\prime}\left(t_{0}\right)=x_{1}$, is the solution given in Problem 7 shifted by an amount $t_{0}$ :
$$
x(t)=x_{0} \cos \omega\left(t-t_{0}\right)+\frac{x_{1}}{\omega} \sin \omega\left(t-t_{0}\right)
$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 9

Find an interval centered about $x=0$ for which the given initial-value problem has a unique solution.
$$
(x-2) y^{\prime \prime}+3 y=x, y(0)=0, y^{\prime}(0)=1
$$

Hast Aggarwal

Numerade Educator

Problem 10

Find an interval centered about $x=0$ for which the given initial-value problem has a unique solution.
$$
y^{\prime \prime}+(\tan x) y=e^{x}, y(0)=1, y^{\prime}(0)=0
$$

Hast Aggarwal

Numerade Educator

Problem 11

(a) Use the family in Problem 1 to find a solution of $y^{\prime \prime}-y=0$ that satisfies the boundary conditions $y(0)=0, y(1)=1$.
(b) The DE in part (a) has the alternative general solution Ir $y=c_{3} \cosh x+c_{4} \sinh x$ on $(-\infty, \infty)$. Use this family to
find a solution that satisfies the boundary conditions in part (a).
(c) Show that the solutions in parts (a) and
(b) are equivalent.

Hast Aggarwal

Numerade Educator

Problem 12

Use the family in Problem 5 to find a solution of $x y^{\prime \prime}-y^{\prime}=0$ that satisfies the boundary conditions $y(0)=1, y^{\prime}(1)=6$.

Hast Aggarwal

Numerade Educator

Problem 13

The given two-parameter family is a solution of the indicated differential equation on the interval $(-\infty, \infty)$. Determine whether a member of the family can be found that satisfies the boundary conditions.
$y=c_{1} e^{x} \cos x+c_{2} e^{x} \sin x ; y^{\prime \prime}-2 y^{\prime}+2 y=0$
(a) $y(0)=1, y^{\prime}(\pi)=0$
(b) $y(0)=1, y(\pi)=-1$
(c) $y(0)=1, y(\pi / 2)=1$
(d) $y(0)=0, y(\pi)=0$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 14

The given two-parameter family is a solution of the indicated differential equation on the interval $(-\infty, \infty)$. Determine whether a member of the family can be found that satisfies the boundary conditions.
$y=c_{1} x^{2}+c_{2} x^{4}+3 ; x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24$
(a) $y(-1)=0, y(1)=4$
(b) $y(0)=1, y(1)=2$
(c) $y(0)=3, y(1)=0$
(d) $y(1)=3, y(2)=15$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 15

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=x, \quad f_{2}(x)=x^{2}, \quad f_{3}(x)=4 x-3 x^{2}
$$

Hast Aggarwal

Numerade Educator

Problem 16

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=0, \quad f_{2}(x)=x, \quad f_{3}(x)=e^{x}
$$

Hast Aggarwal

Numerade Educator

Problem 17

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=5, \quad f_{2}(x)=\cos ^{2} x, \quad f_{3}(x)=\sin ^{2} x
$$

Hast Aggarwal

Numerade Educator

Problem 18

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=\cos 2 x, \quad f_{2}(x)=1, \quad f_{3}(x)=\cos ^{2} x
$$

Hast Aggarwal

Numerade Educator

Problem 19

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=x, \quad f_{2}(x)=x-1, \quad f_{3}(x)=x+3
$$

Hast Aggarwal

Numerade Educator

Problem 20

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=2+x, \quad f_{2}(x)=2+|x|
$$

Hast Aggarwal

Numerade Educator

Problem 21

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=1+x, \quad f_{2}(x)=x, \quad f_{3}(x)=x^{2}
$$

Hast Aggarwal

Numerade Educator

Problem 22

Determine whether the given set of functions is linearly dependent or linearly independent on the interval $(-\infty, \infty)$.
$$
f_{1}(x)=e^{x}, \quad f_{2}(x)=e^{-x}, \quad f_{3}(x)=\sinh x
$$

Hast Aggarwal

Numerade Educator

Problem 23

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
y^{\prime \prime}-y^{\prime}-12 y=0 ; e^{-3 x}, e^{4 x},(-\infty, \infty)
$$

Hast Aggarwal

Numerade Educator

Problem 24

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
y^{\prime \prime}-4 y=0 ; \cosh 2 x, \sinh 2 x,(-\infty, \infty)
$$

Mahnoor Khan

Numerade Educator

Problem 25

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
y^{\prime \prime}-2 y^{\prime}+5 y=0 ; e^{x} \cos 2 x, e^{x} \sin 2 x,(-\infty, \infty)
$$

Hast Aggarwal

Numerade Educator

Problem 26

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
4 y^{\prime \prime}-4 y^{\prime}+y=0 ; e^{\sqrt{2}}, x e^{x / 2},(-\infty, \infty)
$$

Hast Aggarwal

Numerade Educator

Problem 27

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
x^{2} y^{\prime \prime}-6 x y^{\prime}+12 y=0 ; x^{3}, x^{4},(0, \infty)
$$

Hast Aggarwal

Numerade Educator

Problem 28

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
x^{2} y^{\prime \prime}+x y^{\prime}+y=0 ; \cos (\ln x), \sin (\ln x),(0, \infty)
$$

Hast Aggarwal

Numerade Educator

Problem 29

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
x^{3} y^{\prime \prime \prime}+6 x^{2} y^{\prime \prime}+4 x y^{\prime}-4 y=0 ; x, x^{-2}, x^{-2} \ln x,(0, \infty)
$$

Hast Aggarwal

Numerade Educator

Problem 30

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval. Form the general solution.
$$
y^{(4)}+y^{\prime \prime}=0 ; 1, x, \cos x, \sin x,(-\infty, \infty)
$$

Hast Aggarwal

Numerade Educator

Problem 31

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
$$
\begin{aligned}
&y^{\prime \prime}-7 y^{\prime}+10 y=24 e^{x} \\
&y=c_{1} e^{2 x}+c_{2} e^{5 x}+6 e^{x},(-\infty, \infty)
\end{aligned}
$$

Hast Aggarwal

Numerade Educator

Problem 32

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
$$
\begin{aligned}
&y^{\prime \prime}+y=\sec x \\
&y=c_{1} \cos x+c_{2} \sin x+x \sin x+(\cos x) \ln (\cos x) \\
&(-\pi / 2, \pi / 2)
\end{aligned}
$$

Hast Aggarwal

Numerade Educator

Problem 33

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
$$
\begin{aligned}
&y^{\prime \prime}-4 y^{\prime}+4 y=2 e^{2 x}+4 x-12 \\
&y=c_{1} e^{2 x}+c_{2} x e^{2 x}+x^{2} e^{2 x}+x-2,(-\infty, \infty)
\end{aligned}
$$

Hast Aggarwal

Numerade Educator

Problem 34

Verify that the given two-parameter family of functions is the general solution of the nonhomogeneous differential equation on the indicated interval.
$$
\begin{aligned}
&2 x^{2} y^{\prime \prime}+5 x y^{\prime}+y=x^{2}-x \\
&y=c_{1} x^{-1 / 2}+c_{2} x^{-1}+\frac{1}{15} x^{2}-\frac{1}{6} x,(0, \infty)
\end{aligned}
$$

Hast Aggarwal

Numerade Educator

Problem 35

(a) Verify that $y_{P_{1}}=3 e^{2 x}$ and $y_{p_{1}}=x^{2}+3 x$ are, respectively, particular solutions of
$$
y^{\prime \prime}-6 y^{\prime}+5 y=-9 e^{2 x}
$$
and $y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16$
(b) Use part (a) to find particular solutions of
$$
y^{\prime \prime}-6 y^{\prime}+5 y=5 x^{2}+3 x-16-9 e^{2 x}
$$
and $y^{\prime \prime}-6 y^{\prime}+5 y=-10 x^{2}-6 x+32+e^{2 x}$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 36

(a) By inspection, find a particular solution of
$$
y^{n}+2 y=10
$$
(b) By inspection, find a particular solution of
$$
y^{\prime \prime}+2 y=-4 x
$$
(c) Find a particular solution of $y^{\prime \prime}+2 y=-4 x+10$.
(d) Find a particular solution of $y^{\prime \prime}+2 y=8 x+5$.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

Let $n=1,2,3, \ldots$. Discuss how the observations $D^{n} x^{n-1}=0$ and $D^{n} x^{n}=n !$ can be used to find the general solutions of the given differential equations.
(a) $y^{\prime \prime}=0$
(b) $y^{\prime \prime \prime}=0$
(c) $y^{(4)}=0$
(d) $y^{\prime \prime}=2$
(e) $y^{\prime \prime \prime}=6$
(f) $y^{(4)}=24$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 38

Suppose that $y_{1}=e^{x}$ and $y_{2}=e^{-x}$ are two solutions of a homogeneous linear differential equation. Explain why $y_{3}=\cosh x$ and $y_{4}=\sinh x$ are also solutions of the equation.

Mahnoor Khan

Numerade Educator

Problem 39

(a) Verify that $y_{1}=x^{3}$ and $y_{2}=\left\lfloor x^{3}\right.$ are linearly independent solutions of the differential equation $x^{2} y^{\prime \prime}-4 x y^{\prime}+6 y=0$ on the interval $(-\infty, \infty)$.
(b) Show that $W\left(y_{1}, y_{2}\right)=0$ for every real number $x .$ Does this result violate Theorem $3.1 .3$ ? Explain.
(c) Verify that $Y_{1}=x^{3}$ and $Y_{2}=x^{2}$ are also linearly independent solutions of the differential equation in part (a) on the interval $(-\infty, \infty)$.
(d) Find a solution of the differential equation satisfying $y(0)=0, y^{\prime}(0)=0$
(e) By the superposition principle, Theorem $3.1 .2$, both linear combinations $y=c_{1} y_{1}+c_{2} y_{2}$ and $Y=c_{1} Y_{1}+c_{2} Y_{2}$ are solutions of the differentialequation. Discuss whether one, both, or neither of the linear combinations is a general solution of the differential equation on the interval $(-\infty, \infty)$.

Payton Sawyer

Numerade Educator

Problem 40

Is the set of functions $f_{1}(x)=e^{x+2}, f_{2}(x)=e^{x-3}$ linearly dependent or linearly independent on the interval $(-\infty, \infty) ?$ Discuss.

Hast Aggarwal

Numerade Educator

Problem 41

Suppose $y_{1}, y_{2}, \ldots, y_{k}$ are $k$ linearly independent solutions on $(-\infty, \infty)$ of a homogeneous linear $n$ th-order differential equation with constant coefficients. By Theorem $3.1 .2$ it follows that $y_{k+1}=0$ is also a solution of the differential equation. Is the set of solutions $y_{1}, y_{2}, \ldots, y_{b} y_{k+1}$ linearly dependent or linearly independent on $(-\infty, \infty) ?$ Discuss.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 42

Suppose that $y_{1}, y_{2}, \ldots, y_{k}$ are $k$ nontrivial solutions of a homogeneous linear $n$ th-order differential equation with constant coefficients and that $k=n+1$. Is the set of solutions $y_{1}, y_{2}, \ldots, y_{k}$ linearly dependent or linearly independent on $(-\infty, \infty) ?$ Discuss.

Mahnoor Khan

Numerade Educator