Chapter 17: Differential EquationsContent check — Calculus (Guichard), Chapter 17: Differential Equations

Solve ${\displaystyle \dot{y}=1/(1+t^2)}$⁠.

Solve the initial value problem $\dot{y}=t^n$ with $y(0)=1$ and $n\ge 0$⁠.

Solve $\dot{y}=\ln t$⁠.

Identify the constant solutions (if any) of $\dot{y}=t\sin y$⁠.

Identify the constant solutions (if any) of ${\displaystyle \dot{y}=t{e}^y}$⁠.

Solve $\dot{y}=t/y$⁠.

Solve ${\displaystyle \dot{y}=y^2-1}$⁠.

Solve ${\displaystyle \dot{y}=t/(y^3-5)}$⁠. You may leave your solution in implicit form: that is, you may stop once you have done the integration, without solving for $y$⁠.

Find a non-constant solution of the initial value problem $\dot{y}=y^{1/3}$⁠, $y(0)=0$⁠, using separation of variables. Note that the constant function $y(t)=0$ also solves the initial value problem. This shows that an initial value problem can have more than one solution.

Solve the equation for Newton's law of cooling leaving $M$ and $k$ unknown.

After 10 minutes in Jean-Luc's room, his tea has cooled to ${40}^{\circ }$ Celsius from ${100}^{\circ }$ Celsius. The room temperature is ${25}^{\circ }$ Celsius. How much longer will it take to cool to ${35}^{\circ }$⁠?

Solve the logistic equation $\dot{y}=ky(M-y)$⁠. (This is a somewhat more reasonable population model in most cases than the simpler $\dot{y}=ky$⁠.) Sketch the graph of the solution to this equation when $M=1000$⁠, $k=0.002$⁠, $y(0)=1$⁠.

Suppose that $\dot{y}=ky$⁠, $y(0)=2$⁠, and $\dot{y}(0)=3$⁠. What is $y$⁠?

Radioactive substances "decay'', that is, the amount (the mass) of the substance decreases over time. The mass of a radioactive substance obeys the equation $\dot{y}=ky$ where $k<0$ and $y$ is the mass of the substance at time $t$⁠. Suppose that initially, the mass of the substance is $y(0)=M>0$⁠. At what time does half of the mass remain? (This is known as the half life. Note that the half life depends on $k$ but not on $M$⁠.)

Bismuth-210 has a half life of five days. If there is initially 600 milligrams, how much is left after 6 days? When will there be only 2 milligrams left?

The half life of carbon-14 is 5730 years. If one starts with 100 milligrams of carbon-14, how much is left after 6000 years? How long do we have to wait before there is less than 2 milligrams?

A certain species of bacteria doubles its population (or its mass) every hour in the lab. The differential equation that models this phenomenon is $\dot{y}=ky$⁠, where $k>0$ and $y$ is the population of bacteria at time $t$⁠. What is $y$⁠?

If a certain microbe doubles its population every 4 hours and after 5 hours the total population has mass 500 grams, what was the initial mass?

Find the general solution of each equation.

${\displaystyle \dot{y}+5y=0}$

Find the general solution of each equation.

${\displaystyle \dot{y}-2y=0}$

Find the general solution of each equation.

${\displaystyle \dot{y}+\frac{y}{1+t^2}=0}$

Find the general solution of each equation.

${\displaystyle \dot{y}+t^{2}y=0}$

Solve the initial value problem.

${\displaystyle \dot{y}+y=0}$⁠, $y(0)=4$

Solve the initial value problem.

${\displaystyle \dot{y}-3y=0}$⁠, $y(1)=-2$

Solve the initial value problem.

${\displaystyle \dot{y}+y\sin t=0}$⁠, $y(\pi )=1$

Solve the initial value problem.

${\displaystyle \dot{y}+y{e}^t=0}$⁠, $y(0)=e$

Solve the initial value problem.

${\displaystyle \dot{y}+y\sqrt{1+t^4}=0}$⁠, $y(0)=0$

Solve the initial value problem.

${\displaystyle \dot{y}+y\cos (e^t)=0}$⁠, $y(0)=0$

Solve the initial value problem.

${\displaystyle t\dot{y}-2y=0}$⁠, $y(1)=4$

Solve the initial value problem.

${\displaystyle t^2\dot{y}+y=0}$⁠, $y(1)=-2$⁠, $t>0$

Solve the initial value problem.

${\displaystyle t^3\dot{y}=2y}$⁠, $y(1)=1$⁠, $t>0$

Solve the initial value problem.

${\displaystyle t^3\dot{y}=2y}$⁠, $y(1)=0$⁠, $t>0$

A function $y(t)$ is a solution of ${\displaystyle \dot{y}+ky=0}$⁠. Suppose that $y(0)=100$ and $y(2)=4$⁠. Find $k$ and find $y(t)$⁠.

A function $y(t)$ is a solution of ${\displaystyle \dot{y}+t^{k}y=0}$⁠. Suppose that $y(0)=1$ and $y(1)=e^{-13}$⁠. Find $k$ and find $y(t)$⁠.

A bacterial culture grows at a rate proportional to its population. If the population is one million at $t=0$ and 1.5 million at $t=1$ hour, find the population as a function of time.

A radioactive element decays with a half-life of 6 years. If a block of the element has mass 10 kilograms at $t=0$⁠, find the amount of the element at time $t$⁠.

Find the general solution of the equation.

${\displaystyle \dot{y}+4y=8}$

Find the general solution of the equation.

${\displaystyle \dot{y}-2y=6}$

Find the general solution of the equation.

${\displaystyle \dot{y}+ty=5t}$

Find the general solution of the equation.

${\displaystyle \dot{y}+e^{t}y=-2e^t}$

Find the general solution of the equation.

${\displaystyle \dot{y}-y=t^2}$

Find the general solution of the equation.

${\displaystyle 2\dot{y}+y=t}$

Find the general solution of the equation.

${\displaystyle t\dot{y}-2y=1/t}$⁠, $t>0$

Find the general solution of the equation.

${\displaystyle t\dot{y}+y=\sqrt{t}}$⁠, $t>0$

Find the general solution of the equation.

${\displaystyle \dot{y}\cos t+y\sin t=1}$⁠, $-\pi /2<t<\pi /2$

Find the general solution of the equation.

${\displaystyle \dot{y}+y\sec t=\tan t}$⁠, $-\pi /2<t<\pi /2$

Compute the Euler approximations for the initial value problem for $0\le t\le 1$ and $\mathrm{\Delta }t=0.2$⁠.

${\displaystyle \dot{y}=t/y}$⁠, $y(0)=1$

Compute the Euler approximations for the initial value problem for $0\le t\le 1$ and $\mathrm{\Delta }t=0.2$⁠.

${\displaystyle \dot{y}=t+y^3}$⁠, $y(0)=1$

Compute the Euler approximations for the initial value problem for $0\le t\le 1$ and $\mathrm{\Delta }t=0.2$⁠.

${\displaystyle \dot{y}=\cos (t+y)}$⁠, $y(0)=1$

Compute the Euler approximations for the initial value problem for $0\le t\le 1$ and $\mathrm{\Delta }t=0.2$⁠.

${\displaystyle \dot{y}=t\ln y}$⁠, $y(0)=2$

Solve the initial value problem ${\displaystyle \ddot{y}-{\omega }^{2}y=0}$⁠, $y(0)=1$⁠, ${\displaystyle \dot{y}(0)=1}$⁠, assuming $\omega ≠0$⁠.

Solve the initial value problem ${\displaystyle 2\ddot{y}+18y=0}$⁠, $y(0)=2$⁠, ${\displaystyle \dot{y}(0)=15}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}+6\dot{y}+5y=0}$⁠, $y(0)=1$⁠, ${\displaystyle \dot{y}(0)=0}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}-\dot{y}-12y=0}$⁠, $y(0)=0$⁠, ${\displaystyle \dot{y}(0)=14}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}+12\dot{y}+36y=0}$⁠, $y(0)=5$⁠, ${\displaystyle \dot{y}(0)=-10}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}-8\dot{y}+16y=0}$⁠, $y(0)=-3$⁠, ${\displaystyle \dot{y}(0)=4}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}+5y=0}$⁠, $y(0)=-2$⁠, ${\displaystyle \dot{y}(0)=5}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}+y=0}$⁠, $y(\pi /4)=0$⁠, ${\displaystyle \dot{y}(\pi /4)=2}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}+12\dot{y}+37y=0}$⁠, $y(0)=4$⁠, ${\displaystyle \dot{y}(0)=0}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}+6\dot{y}+18y=0}$⁠, $y(0)=0$⁠, ${\displaystyle \dot{y}(0)=6}$⁠.

Solve the initial value problem ${\displaystyle \ddot{y}+4y=0}$⁠, $y(0)=\sqrt{3}$⁠, ${\displaystyle \dot{y}(0)=2}$⁠. Put your answer in the form developed at the end of example 17.5.3.

Solve the initial value problem ${\displaystyle \ddot{y}+100y=0}$⁠, $y(0)=5$⁠, ${\displaystyle \dot{y}(0)=50}$⁠. Put your answer in the form developed at the end of example 17.5.3.

Solve the initial value problem ${\displaystyle \ddot{y}+4\dot{y}+13y=0}$⁠, $y(0)=1$⁠, ${\displaystyle \dot{y}(0)=1}$⁠. Put your answer in the form developed at the end of example 17.5.3.

Solve the initial value problem ${\displaystyle \ddot{y}-8\dot{y}+25y=0}$⁠, $y(0)=3$⁠, ${\displaystyle \dot{y}(0)=0}$⁠. Put your answer in the form developed at the end of example 17.5.3.

A mass-spring system ${\displaystyle m\ddot{y}+b\dot{y}+ky}$ has $k=29$⁠, $b=4$⁠, and $m=1$⁠. At time $t=0$ the position is $y(0)=2$ and the velocity is $\dot{y}(0)=1$⁠. Find $y(t)$⁠.

A mass-spring system ${\displaystyle m\ddot{y}+b\dot{y}+ky}$ has $k=24$⁠, $b=12$⁠, and $m=3$⁠. At time $t=0$ the position is $y(0)=0$ and the velocity is $\dot{y}(0)=-1$⁠. Find $y(t)$⁠.

Find the general solution to the differential equation.

${\displaystyle \ddot{y}-10\dot{y}+25y=\cos t}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+2\sqrt{2}\dot{y}+2y=10}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+16y=8t^2+3t-4}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+2y=\cos (5t)+\sin (5t)}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}-2\dot{y}+2y=e^{2t}}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}-6y+13=1+2t+e^{-t}}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+\dot{y}-6y=e^{-3t}}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}-4\dot{y}+3y=e^{3t}}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+16y=\cos (4t)}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+9y=3\sin (3t)}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+12\dot{y}+36y=6e^{-6t}}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}-8\dot{y}+16y=-2e^{4t}}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+6\dot{y}+5y=4}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}-\dot{y}-12y=t}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}+5y=8\sin (2t)}$

Find the general solution to the differential equation.

${\displaystyle \ddot{y}-4y=4e^{2t}}$

Solve the initial value problem.

${\displaystyle \ddot{y}-y=3t+5}$⁠, $y(0)=0$⁠, ${\displaystyle \dot{y}(0)=0}$

Solve the initial value problem.

${\displaystyle \ddot{y}+9y=4t}$⁠, $y(0)=0$⁠, ${\displaystyle \dot{y}(0)=0}$

Solve the initial value problem.

${\displaystyle \ddot{y}+12\dot{y}+37y=10e^{-4t}}$⁠, $y(0)=4$⁠, ${\displaystyle \dot{y}(0)=0}$

Solve the initial value problem.

${\displaystyle \ddot{y}+6\dot{y}+18y=\cos t-\sin t}$⁠, $y(0)=0$⁠, ${\displaystyle \dot{y}(0)=2}$

Solve the initial value problem.

Find the solution for the mass-spring equation ${\displaystyle \ddot{y}+4\dot{y}+29y=689\cos (2t)}$⁠.

Solve the initial value problem.

Find the solution for the mass-spring equation ${\displaystyle 3\ddot{y}+12\dot{y}+24y=2\sin t}$⁠.

Find the general solution to the differential equation using variation of parameters.

${\displaystyle \ddot{y}+y=\tan t}$

Find the general solution to the differential equation using variation of parameters.

${\displaystyle \ddot{y}+y=e^{2t}}$

Find the general solution to the differential equation using variation of parameters.

${\displaystyle \ddot{y}+4y=\sec t}$

Find the general solution to the differential equation using variation of parameters.

${\displaystyle \ddot{y}+4y=\tan t>}$

Find the general solution to the differential equation using variation of parameters.

${\displaystyle \ddot{y}+\dot{y}-6y=t^2{e}^{2t}}$

Find the general solution to the differential equation using variation of parameters.

${\displaystyle \ddot{y}-2\dot{y}+2y=e^t\tan (t)}$

Find the general solution to the differential equation using variation of parameters.

${\displaystyle \ddot{y}-}$ ${\displaystyle 2\dot{y}+}$ ${\displaystyle 2y=}$ ${\displaystyle \sin (t)\cos (t)}$ (This is rather messy when done by variation of parameters; compare to undetermined coefficients.)