Homework 1 - Introduction and Direction Fields

Quiz 1 Practice · Lecture 1 - Introduction · Homework 2

Assigned order from the course website. Problem statements are condensed from the textbook scans; numbering and mathematical data are retained.

Section 1.2

5

Find every constant $k$ for which $y=e^{kt}$ satisfies $y^{\prime }+2y=0$.

7

Find every constant $k$ for which $y=e^{k\cos 2t}$ satisfies $y^{\prime }+(\sin 2t)y=0$.

8

Find every constant $k$ for which $y=k{e}^{-t}$ satisfies $y^{\prime }+y=0$.

9

Verify $y=C{e}^{t^2}$ solves $y^{\prime }-2ty=0$ for arbitrary $C$. Then impose $y(1)=2$.

13

For $t>0$, find every $c$ such that $y=c/t$ solves $y^{\prime }+y^2=0$.

14

Given $y=-e^{-t}+\sin t$, recover $g(t)$ and $y_0$ in

$$y^{\prime }+y=g(t),y(0)=y_0.$$

21

The solution of $y^{\prime }=m+1$, $y(1)=y_0$ is the line shown. Determine the integer $m$, $y_0$, and $y(t)$.

23

An object is released from rest at height $y_0>0$. With constant gravitational acceleration $g>0$, derive the impact time and impact velocity in terms of $y_0,g$.

Instructor hint

Start with $y^{\prime \prime }=-g$, determine both initial conditions, and integrate twice as in Lecture 1. Count the arbitrary constants before imposing initial data. Impact occurs when $y(t)=0$.

3

Determine the order:

$$(y^{\prime }{)}^3+t^5\sin y=y^4.$$

10

Solve $y^{\prime \prime \prime }=2$ by successive antiderivatives. State its order and count the arbitrary constants.

Section 1.3

For 5, 1, 2, 4, determine whether the equation is autonomous, find all equilibria, and sketch its direction field on $-2\le t\le 2$, $-2\le y\le 2$.

5

$$y^{\prime }=-1.$$

1

$$y^{\prime }=-y+1.$$

Additionally sketch the solution through $y(0)=1/3$.

2

$$y^{\prime }=t-1.$$

4

$$y^{\prime }=y^2-y.$$

Additionally sketch the solution through $y(0)=1/3$.

Instructor guidance for 1 and 4

The original HW 1 instructions ask for Mathematica DSolve to obtain the two IVP curves, then a check against the direction fields. For #4, locate the equilibria first. For hand practice at the Quiz 1 cutoff, sketch the nonlinear trajectory qualitatively; separation of variables is introduced later. #1 can now be solved by Lecture 2’s constant-shift method.

10

Construct an autonomous DE with equilibria $y=0,2$, positive slope for $0<y<2$, and negative slope for $y<0$ or $y>2$.

Sources and answer checks

Assignment and instructor hints, checked September 9, 2026.

Textbook: Kohler and Johnson, Elementary Differential Equations, 2nd ed., §§1.2–1.3. Local scans: MATH3230-HW-sec1.2.pdf, MATH3230-HW-sec1.3.pdf.

Posted answers: MATH3230-HW-answers-sec1.2-1.3.pdf. Additional even-numbered answers appear on the assignment webpage.