Elementary Differential Equations with Boundary Value Problems

Werner E. Kohler, Lee W. Johnson

Chapter 2

First Order Differential Equations - all with Video Answers

Educators

Section 1

Introduction

Problem 1

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
y^{\prime}-\sin t=t^{2} y
$$

Amy Jiang

Amy Jiang

Numerade Educator

Problem 2

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
y^{\prime}-\sin t=t y^{2}
$$

Amy Jiang

Numerade Educator

Problem 3

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
\frac{y^{\prime}}{y}-y \cos t=t
$$

Amy Jiang

Numerade Educator

Problem 4

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
y^{\prime} \sin y=\left(t^{2}+1\right) y
$$

Albert T.

Numerade Educator

Problem 5

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
y^{\prime} \sin t=\frac{t^{2}+1}{y}
$$

Amy Jiang

Numerade Educator

Problem 6

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
2 t y+e^{t} y^{\prime}=\frac{y}{t^{2}+4}
$$

Amy Jiang

Numerade Educator

Problem 7

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
y y^{\prime}=t^{3}+y \sin 3 t
$$

Amy Jiang

Numerade Educator

Problem 8

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
2 t y+e^{y} y^{\prime}=\frac{y}{t^{2}+4}
$$

Amy Jiang

Numerade Educator

Problem 9

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
\frac{t y^{\prime}}{\left(t^{4}+2\right) y}=\cos t+\frac{e^{3 t}}{y}
$$

Amy Jiang

Numerade Educator

Problem 10

Classify each of the following first order differential equations as linear or nonlinear. If the equation is linear, decide whether it is homogeneous or nonhomogeneous.
$$
\frac{y^{\prime}}{\left(t^{2}+1\right) y}=\cos t
$$

Amy Jiang

Numerade Educator

Problem 11

Consider the following first order linear differential equations. For each of the initial conditions, determine the largest interval $a<t<b$ on which Theorem $2.1$ guarantees the existence of a unique solution.
$y^{\prime}+\frac{t}{t^{2}+1} y=\sin t$
(a) $y(-2)=1$
(b) $y(0)=\pi$
(c) $y(\pi)=0$

Monica Miller

Numerade Educator

Problem 12

Consider the following first order linear differential equations. For each of the initial conditions, determine the largest interval $a<t<b$ on which Theorem $2.1$ guarantees the existence of a unique solution.
$$
y^{\prime}+\frac{t}{t^{2}-4} y=0
$$
(a) $y(6)=2$
(b) $y(1)=-1$
(c) $y(0)=1$
(d) $y(-6)=2$

Monica Miller

Numerade Educator

Problem 13

Consider the following first order linear differential equations. For each of the initial conditions, determine the largest interval $a<t<b$ on which Theorem $2.1$ guarantees the existence of a unique solution.
$y^{\prime}+\frac{t}{t^{2}-4} y=\frac{e^{t}}{t-3}$
(a) $y(5)=2$
(b) $y\left(-\frac{3}{2}\right)=1$
(c) $y(0)=0$
(d) $y(-5)=4$
(e) $y\left(\frac{3}{2}\right)=3$

Monica Miller

Numerade Educator

Problem 14

Consider the following first order linear differential equations. For each of the initial conditions, determine the largest interval $a<t<b$ on which Theorem $2.1$ guarantees the existence of a unique solution.
$y^{\prime}+(t-1) y=\frac{\ln \left|t+t^{-1}\right|}{t-2}$
(a) $y(3)=0$
(b) $y\left(\frac{1}{2}\right)=-1$
(c) $y\left(-\frac{1}{2}\right)=1$
(d) $y(-3)=2$

Monica Miller

Numerade Educator

Problem 15

If $y(t)=3 e^{t^{2}}$ is known to be the solution of the initial value problem
$$
y^{\prime}+p(t) y=0, \quad y(0)=y_{0},
$$
what must the function $p(t)$ and the constant $y_{0}$ be?

Sriram Soundarrajan

Sriram Soundarrajan

Numerade Educator

Problem 16

(a) For what value of the constant $C$ and exponent $r$ is $y=C t^{r}$ the solution of the initial value problem
$$
2 t y^{\prime}-6 y=0, \quad y(-2)=8 ?
$$
(b) Determine the largest interval of the form $(a, b)$ on which Theorem $2.1$ guarantees the existence of a unique solution.
(c) What is the actual interval of existence for the solution found in part (a)?

Alexandra Embry

Alexandra Embry

Numerade Educator

Problem 17

If $p(t)$ is any function continuous on an interval of the form $a<t<b$ and if $t_{0}$ is any point lying within this interval, what is the unique solution of the initial value problem
$$
y^{\prime}+p(t) y=0, \quad y\left(t_{0}\right)=0
$$
on this interval? [Hint: If, by inspection, you can identify one solution of the given initial value problem, then Theorem $2.1$ tells you that it must be the only solution.]

Aidan Mcnabb

Numerade Educator