Chapter 2: First-order equationsContent check — Differential Equations (Lebl), Chapter 2: First-order equations

Solve $\frac{dy}{dx}=e^x+x$ and $y(0)=10$⁠.

Solve $x^{\prime }=\frac{1}{x^2}$⁠, $x(1)=1$⁠.

Solve $x^{\prime }=\frac{1}{\cos (x)}$⁠, $x(0)=\frac{\pi }{4}$⁠.

Sid is in a car traveling at speed $10t+70$ miles per hour away from Las Vegas, where $t$ is in hours. At $t=0$⁠, Sid is 10 miles away from Vegas. How far from Vegas is Sid 2 hours later?

Solve $y^{\prime }=y^n$⁠, $y(0)=1$⁠, where $n$ is a positive integer. Hint: You have to consider different cases.

The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical. The volume (in centimeters cubed) of a ball of radius $r$ centimeters is $({}^4/{}_3)\pi r^3$⁠. The surface area is $4\pi r^2$⁠. Set up the differential equation for how the radius $r$ is changing. Then, suppose that at time $t=0$ minutes, the radius is 10 centimeters. After 5 minutes, the radius is 8 centimeters. At what time $t$ will the snowball be completely melted?

Find the general solution to $y^{\prime \prime \prime \prime }=0$⁠. How many distinct constants do you need?

Sketch the slope field of $y^{\prime }=y^3$⁠. Can you visually find the solution that satisfies $y(0)=0$⁠?

Is it possible to solve $y^{\prime }=xy$ for $y(0)=0$⁠? Is the solution unique?

Is it possible to solve $y^{\prime }=\frac{x}{x^2-1}$ for $y(1)=0$⁠?

Match equations $y^{\prime }=\sin x$⁠, $y^{\prime }=\cos y$⁠, $y^{\prime }=y\cos (x)$ to slope fields. Justify.

a)

(tricky) Suppose

$$f(y)=\{\begin{array}{cc}0\hfill & \text{ if }y>0,\hfill \\ 1\hfill & \text{ if }y\le 0.\hfill \end{array}$$

Does $y^{\prime }=f(y)$⁠, $y(0)=0$ have a continuously differentiable solution? Does Picard apply? Why, or why not?

Consider an equation of the form $y^{\prime }=f(x)$ for some continuous function $f$⁠, and an initial condition $y(x_0)=y_0$⁠. Does a solution exist for all $x$⁠? Why or why not?

Solve $y^{\prime }=2xy$⁠.

Solve $x^{\prime }=3x{t}^2-3t^2$⁠, $x(0)=2$⁠.

Find an implicit solution for $x^{\prime }={\displaystyle \frac{1}{3x^2+1}}$⁠, $x(0)=1$⁠.

Find an explicit solution to $x{y}^{\prime }=y^2$⁠, $y(1)=1$⁠.

Find an implicit solution to $y^{\prime }={\displaystyle \frac{\sin (x)}{\cos (y)}}$⁠.

Take Example 1.3.3 with the same numbers: 89 degrees at $t=0$⁠, 85 degrees at $t=1$⁠, and ambient temperature of 22 degrees. Suppose these temperatures were measured with a precision of $\pm 0.5$ degrees. Given this imprecision, the time it takes the coffee to cool to (exactly) 60 degrees is also only known in a certain range. Find this range. Hint: Think about what kind of error makes the cooling time longer and what shorter.

A population $x$ of rabbits on an island is modeled by $x^{\prime }=x-\left({}^1/{}_{1000}\right)x^2$⁠, where the independent variable $t$ is time in months. At time $t=0$⁠, there are 40 rabbits on the island.

a) Find the solution to the equation with the initial condition.

b) How many rabbits are on the island in 1 month, 5 months, 10 months, 15 months (round to the nearest integer)?

Solve $y^{\prime }+3x^{2}y=x^2$⁠.

Solve $y^{\prime }+2\sin (2x)y=2\sin (2x)$⁠, $y({}^{\pi }/{}_2)=3$⁠.

Suppose a water tank is being pumped out at $3\frac{\mathrm{L}}{\min }$. The water tank starts at $10\mathrm{L}$ of clean water. Water with a toxic substance is flowing into the tank at $2\frac{\mathrm{L}}{\min }$, with concentration $20t\frac{\mathrm{g}}{\mathrm{L}}$ at time $t$⁠. When the tank is half empty, how many grams of toxic substance are in the tank (assuming perfect mixing)?

There are bacteria on a plate and a toxic substance is being added that slows down the rate of growth of the bacteria. That is, suppose that $\frac{dP}{dt}=(2-0.1t)P$⁠. If $P(0)=1000$⁠, find the population at $t=5$⁠.

A cylindrical water tank has water flowing in at $I$ cubic meters per second. Let $A$ be the area of the cross section of the tank in square meters. Suppose water is flowing out from the bottom of the tank at a rate proportional to the height of the water level. Set up the differential equation for $h$⁠, the height of the water, introducing and naming constants that you need. You should also give the units for your constants.

Solve $x{y}^{\prime }+y+y^2=0$⁠, $y(1)=2$⁠.

Solve $x{y}^{\prime }+y+x=0$⁠, $y(1)=1$⁠.

Solve $y^2{y}^{\prime }=y^3-3x$⁠, $y(0)=2$⁠.

Solve $2y{y}^{\prime }=e^{y^2-x^2}+2x$⁠.

Let $x^{\prime }=(x-1)(x-2)x^2$⁠.

a) Sketch the phase diagram and find critical points.

b) Classify the critical points.

c) If $x(0)=0.5$⁠, then find ${\displaystyle \underset{t\to \infty }{\lim }x(t)}$⁠.

Let $x^{\prime }=e^{-x}$⁠.

a) Find and classify all critical points.

b) Find ${\displaystyle \underset{t\to \infty }{\lim }x(t)}$ given any initial condition.

Assume that a population of fish in a lake satisfies $\frac{dx}{dt}=kx(M-x)$⁠. Now suppose that fish are continually added at $A$ fish per unit of time.

a) Find the differential equation for $x$⁠.

b) What is the new limiting population?

Suppose $\frac{dx}{dt}=(x-\alpha )(x-\beta )$ for two numbers $\alpha <\beta$⁠.

a) Find the critical points, and classify them.

For b), c), d), find ${\displaystyle \underset{t\to \infty }{\lim }x(t)}$ based on the phase diagram.

a) $x(0)<\alpha$⁠,

b) $\alpha <x(0)<\beta$⁠,

c) $\beta <x(0)$⁠.

Let $x^{\prime }=\sin (xt)$⁠, and $x(0)=1$⁠. Approximate $x(1)$ using Euler’s method with step sizes 1, 0.5, 0.25. Use a calculator and compute up to 4 decimal digits.

Let $x^{\prime }=2t$⁠, and $x(0)=0$⁠.

a) Approximate $x(4)$ using Euler’s method with step sizes 4, 2, and 1.

b) Solve exactly, and compute the errors.

c) Compute the factor by which the errors changed.

Let $x^{\prime }=x{e}^{xt+1}$⁠, and $x(0)=0$⁠.

a) Approximate $x(4)$ using Euler’s method with step sizes 4, 2, and 1.

b) Guess an exact solution based on part a) and compute the errors.

Consider ${\displaystyle \frac{dy}{dx}}=x+y$⁠, $y(0)=1$⁠.

a) Approximate $y(1)$ using the improved Euler’s method (see above) with step sizes $h={}^1/{}_4$ and $h={}^1/{}_8$⁠.

b) Approximate $y(1)$ using Euler’s method with $h={}^1/{}_4$ and $h={}^1/{}_8$⁠.

c) Solve exactly, find the exact value of $y(1)$⁠.

d) Compute the errors, and the factors by which the errors changed.

Solve the following exact equations, implicit general solutions will suffice:

a) $\cos (x)+y{e}^{xy}+x{e}^{xy}{y}^{\prime }=0$

b) $(2x+y)dx+(x-4y)dy=0$

c) $e^x+e^y\frac{dy}{dx}=0$

d) $(3x^2+3y)dx+(3y^2+3x)dy=0$

Find the integrating factor for the following equations making them into exact equations:

a) $\frac{1}{y}dx+3ydy=0$

b) $dx-e^{-x-y}dy=0$

c) $(\frac{\cos (x)}{y^2}+\frac{1}{y})dx+$ $\frac{x}{y^2}dy=$ $0$

d) $(2y+\frac{y^2}{x})dx+(2y+x)dy=0$

a) Show that every separable equation $y^{\prime }=f(x)g(y)$ can be written as an exact equation, and verify that it is indeed exact.

b) Rewrite $y^{\prime }=xy$ as an exact equation, solve it, and verify that the solution is the same as it was in Example 1.3.1.

Solve

a) $-5u_x+u_t=0$⁠,   $u(x,0)=\frac{1}{1+x^2}$⁠,

b) $2u_x+u_t=0$⁠,   $u(x,0)=\cos (x)$⁠.

Solve $u_x+u_t+tu=0$⁠, $u(x,0)=\cos (x)$⁠.

Solve $u_x+u_t=5$⁠, $u(x,0)=x$⁠.