In this problem, please write down examples of the Differential Equation that are
DO NOT SOLVE THESE EQUATIONS
In the following equations circle all properties that are true
ODE ,PDE , Linear , Nonlinear, First Order , Second Order, Third Order, Homogeneous, Non-homogeneous, Single Equation, System of Equations
In this problem, please write down examples of the Differential Equation that are
DO NOT SOLVE THESE EQUATIONS
During the darkest period of the History of Magic, the Death Eaters movement of followers of Lord Voldemort was gaining momentum fast. The rate of growth of a number of Death Eaters is proportional to the amount of Death Eaters. In the absence of other factors, the number of Death Eaters tripples every two days. One Death Eater is converted to the Order of the Phoenix every day (i.e. stops being Death Eater). Assuming that at time there were only five Death Eaters, calculate the amount of Death Eaters in a week.
Extra Credit 10 percent What happens with the number of the Death Eaters as ? If the number of Death Eaters becomes too large, please modify the equation to give the Order of the Phoenix (the good guys) the chance of victory.
Ten years ago, an industrial company built a factory in a pristine valley. The valley's volume is . The factory started spewing of pollutants through smoke stacks with volume flow . Well-mixed polluted air leaves the valley at the same rate. What is the concentration of the pollutants in the valley now?
HINT: Use the approximate value 1/3 for .
The volume of a lake oscillates yearly as , where = year, and is measured from January 1, 2001. A creek with flow rate flows into the lake. On January 1, 2001, a new farm began operating near this creek and spilling refuse into it. The concentration of this refuse in the creek water is . Well-mixed polluted water leaves the lake at the same rate . Write down the initial-value problem that describes the pogllution level, i.e., mass of the refuse, , in the lake. Do not solve it.
The speed at which DFQ-pen gains popularity and is adopted by RPI students is proportional to the percent of students having the DFQ-pen and the percent of students not having the smart DFQ-pen. Assume that coefficient of proportionality is such that at the beginning, when almost nobody uses the smart DFQ pen, the number of people having the smart DFQ pen triples every week. Find the number of students having the smart DFQ pen as a function of time.
Extra Credit 10 percent Assuming that initially only one person
out of a hundred have the smart DFQ pen, estimate number of people
using the pen in 28 days. To find a final answer, you may approximate
the value of
HINT: You must first compute Jane's salary at any given moment.
The number of RPI students wearing blue jeans change in time
according to the equation
The number of the members of the Order of Phoenix changes
according to the law
Hint The raise of numbers of the Order of Phoenix members are favorable to the order of Phoenix mission, decline of numbers of members of Order of Phoenix is favorable to Death Eaters.
The speed of propagation of gossip in a closed group is directly proportional to a percentage of people knowing the gossip and is also proportional the percentage of people not knowing the gossip. Assume that coefficient of proportionality is such that at the beginning of propagation of a gossip, when almost nobody knows the gossip, the number of people knowing the gossip tripples every two days. Find the number of people knowing the gossip as a function of time.
Extra Credit 10 percent Assuming that initially gossip is known
by one person out of a hundred, estimate a fraction of people knowing
the gossip in a week. To find a final answer, you may approximate the
value of
Steps to be performed to solve the problem are
HINTS:
Extra Credit 10 % What happens as ? Propose the mechanism which leads to more realistic predictions, and how would you change the equation to model this mechanism?
A population of squirrels of the Prospect Park
in Troy NY is limited by the finate area of the part,
changes according to the differential equation
We have studied the integrating factor method to solve
Now suppose that function is such that
(7) |
Explain and resolve this apparent contradiction
Consider the first order ordinary linear ordinary differential
equation
Solve the initial-value problem
Extra Credit: Sketch sufficiently many representative integral curves of the differential equation in () to show how all its solutions behave.
Find the explicit solution of the initial-value problem
Extra credit: Sketch this solution.
Find the explicit solution of the initial-value problem
Solve the initial value problem
Find the explicit solution of
the initial-value problem
Solve the initial-value problem
The number of bacteria in a Petri dish is governed by
the
equation
Consider the problem
HINT: If you have trouble with the abstract constant , do it for , , and .
HINT: If you have trouble with the abstract constant, , , and will do the trick.
Calculate the general solution of the equation
Calculate the solution to the initial value problem
Compute the solution of the initial value problem
Compute the solution of the initial value problem
Find the FORM of the general solution of the problem
Calculate the general solution of the equation
HINT: Think carefully what method you will use for computing the particular solution of the inhomogeneous problem!
HINT:
Find the general solution of the equation
(a) Compute the general solution of the problem
(b) Find the FORM of the general solution of the problem
Solve
Sketch .
Find the general solution of the following first order ODE:
This is nonlinear ODE, so the method we studied in class can not be used directly. Yet, you can find an integrating factor, i.e. the function which can be used to multiply both Left Hand Side and Right Hand Side of this equation to make the Right Hand Side become a full derivative. You may try to use your ``regular'' integrating factor method and see what happens.
If this fails, there exists a change of variables which makes this equation linear.
Extra credit: Sketch this solution.
extra credit 20% What is the general solution of the equation
4. Consider the problem
HINT: If you have trouble with the abstract constant , do it for , , and .
Extra Credit 20 percent
We have studied integrating factor
method for solving general first order linear ODE. Integrating
factor method can sometimes be generalized for nonlinear first order
ODE's.
Find the general solution of the following first order ODE:
This is nonlinear ODE, so the method we studied in class can not be used directly. Yet, you can find an integrating factor, i.e. the function which can be used to multiply both Left Hand Side and Right Hand Side of this equation to make the Right Hand Side become a full derivative. You may try to use your ``regular'' integrating factor method and see what happens.
If this fails, there exists a change of variables which makes this equation linear.
Solve the initial value problem
Find the general solution to the equation
Solve the inhomogeneous ordinary differential equation of a second order
by method of undetermined coefficients:
Solve
Solve Initial Value Problem
Solve
Find the general solution of the equation
Find the general solution of the equation
Solve the following
inhomogeneous ordinary differential equation of a second order
Note that the
general solution to the corresponding homogeneous equation
Calculate the general solution of the equation
We have studied in class the equation
Show how to solve equation () by method of variation of a parameter. To do this, complete the following steps:
A spring/mass system is modeled by the
initial value problem:
(a) Give a qualitatively accurate sketch of for each
indicated value of :
(c) Extra Credit 5 percent
Give a qualitatively accurate sketch of for
If an undamped spring-mass system with the mass that weights 6 lb and a spring constant 1 lb/inch is suddenly set in motion from a rest position at an equilibrium at by an external force of lb, determine the position of the mass at any time and scetch a graph of the displacement versus time.
A vibrating spring-mass system has mass , damping coefficient , and spring stiffness . At time , when the mass is at rest, the force starts acting on it. What is the motion for all subsequent times, ? What motion does the system settle into for large ? Does this large-time motion depend on the initial conditions? Sketch this large-time motion.
Consider the equation for the Damped Harmonic Oscillator:
(b) If now the system is forced so
that it becomes
(a) A vibrating mass-spring system is described by the equation
(b) Suppose that the vibrating system of part (a) is forced so that
Please write down the examples of the equations that corresponds to these cases 1,2,3,4,5 and scetch the possible graphs of for each of these cases. You do not have to solve these equations to scetch these graphs.
Some forced mass-spring system is described by the equation
(i) For , find the solution of () that satisfies the initial conditions , .
(ii) After an initial transient period, every solution of () settles into a stable periodic motion. Identify this motion, and compute its amplitude, A. For what is the largest?
(iii) Roughly sketch the solution that you have obtained in (i) if you assume that .
Find the solution of the initial value problem
Treat each time interval separately and match the solutions in the different intervals requiring and are continuous functions of time.
A spring mass system has a spring constant of 3 Newtons per meter. A mass of 2 kg is attached to the spring, and the motion takes place in a viscous fluid that offers a resistance numerically equal to the magnitude of the instantaneous velocity (i.e. ). If the system is driven by an external force of , ^{1} determine the steady state response of the system (i.e. the form of the solution when ). Express your answer as in the form . Time is measured in seconds.
Some differential equation
has two
linearly independent solutions and . What is the general solution of the equation
Suppose that is even function, and that is odd function. Prove that is odd, and that is also odd. Finally, proove that is neither odd nor even.
Let the function be given in the interval
by the formulas
Find the Full Fourier Series of a Function , defined for
as
Calculate explicitly coefficients for , and .
Sketch the graph of the function to which the series converges for .
Find the Full Fourier Series of a Function , defined for
as
Calculate explicitly coefficients for , and .
Sketch the graph of the function to which the series converges for .
Let the function be given in the interval
by the formulas
(a) Find the Fourier sine and cosine series of the function
(b) Sketch the functions to which the two series of part (a) converge. What is the period of those two functions?
(a) Find the Fourier sine and cosine series of the function
(b) Sketch the functions to which the two series of part (a) converge. What is the period of those two functions? Are these functions even or odd?
(a) Find the Fourier series of
(b) Sketch the graph of the function to which the series converges for three periods.
Find the Fourier Cosine Series for the function for .
Find the Full Fourier Series for the function for and plot the graph of the function to which the series converge for
(a) Find the solution of the eigenvalue problem
(b) Find all the solutions that satisfy the heat
equation
(a) Find the solution of the eigenvalue problem
(b) Find all the solutions that satisfy the heat
equation
Find all the eigenvalues and eigenfunctions of the problem
NOTE: If you use the ``rule of thumb'' mentioned in class, state that explicitly and explain what this rule is.
NOTE: Consider separately , and cases.
Draw (on the same set of axes) the first three eigenfunctions.
Solve
HINT: first solve the ODE (), its solution has two arbitrary constants. Use boundary conditions () to find these two arbitrary constants.
Find all the eigenvalues and eigenfunctions of the problem
NOTE: Please consider explicitly positive eigenvalues, zero eigenvalues and negative eigenvalues.
Find the solution of the eigenvalue problem
NOTE: Please be careful to consider both positive, zero, and negative eigenvalues, or give arguments supporting your choice of sign of the eigenvalues.
Find the eigenvalues and eigenfunctions of the problem
(a) Let . Assuming that , find ordinary differential equations satisfied by and .
(b) bonus 10 percent
Find the steady state of the problem
Given that the eigenvalues and eigenfunctions for the
problem
(a) Let . Assuming that , find ordinary differential equations satisfied by and .
(b) bonus 10 percent
Find the steady state of the problem
Sketch for three values of .
NOTE: Please be careful to consider both positive, zero, and negative eigenvalues, or give arguments supporting your choice of sign of the eigenvalues.
By using the method of separation of variables replace the partial
differential equation
Use separation of variables to find the function
that satisfies the wave equation
Sketch for three values of .
Use separation of variables to replace partial differential equation
Find the general solution of the given system of equations. Sketch a number of representative trajectories.
(a) Consider the heat conduction problem
Dr Yuri V Lvov 2016-08-26