Question

1) State the main difference between an ODE and a PDE? 2) Name two of the...


1) State the main difference between an ODE and a PDE?


2) Name two of the three archetypal PDEs?


3) Write the equation used to compute the Wronskian for two differentiable
functions, y1 and y2.


4) What can you conclude about two differentiable functions, y1 and y2, if their
Wronskian is nonzero?
5) (2 pts) If two functions, y1 and y2, solve a 2nd order DE, what does the Principle of
Superposition guarantee?


6) (8 pts, 4 pts each) State the order of each ODE, then classify each of them as
linear/nonlinear, homogeneous/inhomogeneous, and autonomous/nonautonomous.
A) Unforced Pendulum: θ′′ + γ θ′ + ω^2sin θ = 0


B) Simple RLC Circuit with a 9V Battery: Lq′′ + Rq′ +(1/c)q = 9


7) (8 pts) Find all critical points for the given DE, draw a phase line for the system,
then state the stability of each critical point.
Logistic Equation: y′ = ry(1 − y/K), where r < 0


8) (6 pts) A mass of 2 kg is attached to the end of a spring and is acted on by an
external, driving force of 8 sin(t) N. When in motion, it moves through a medium that
imparts a viscous force of 4 N when the speed of the mass is 0.1 m/s. The spring
constant is given as 3 N/m, and this mass-spring system is set into motion from its
equilibrium position with a downward initial velocity of 1 m/s. Formulate the IVP
describing the motion of the mass. DO NOT SOLVE THE IVP.


9) (8 pts, 4 pts each) Find the maximal interval of existence, I, for each IVP given.
A) (t^2 − 9) y′ − 7t^3 =√t, y(−2) = 12


B) sin(t) y′′ + ty′ − 18y = 1, y(4) = 9, y′(4) = −13


10) (30 pts, 10 pts each) Solve for the general solution to each of the DEs given. Use an
appropriate method in each case.
A) Newton’s Law of Cooling: y′ = −k(y − T)


B) (sin(y) − y sin(t)) dt + (cos(t) + t cos(y) − y) dy = 0


C) ty′ − 5y = t^6 *e^t

11) (30 pts, 10 pts each) Solve for the general solution to each of the DEs given, then
classify the stability and type of critical point that lies at the origin for each case.
A) y′′ + y′ − 132y = 0


B) y′′ + 361y = 0


C) y′′ + 6y′ + 10y = 0

12) (10 pts) Solve for the general solution to the DE given.
y′′ − 9y = −18t^2 + 6


13) (15 pts) Solve the given IVP.
y′′ + 2y′ + 2y = 10 sin(2t), y(0) = 1, y′(0) = 0

Homework Answers

Know the answer?
Your Answer:

Post as a guest

Your Name:

What's your source?

Earn Coins

Coins can be redeemed for fabulous gifts.

Not the answer you're looking for?
Ask your own homework help question
Similar Questions
6) (8 pts, 4 pts each) State the order of each ODE, then classify each of...
6) (8 pts, 4 pts each) State the order of each ODE, then classify each of them as linear/nonlinear, homogeneous/inhomogeneous, and autonomous/nonautonomous. A) Unforced Pendulum: θ′′ + γ θ′ + ω^2sin θ = 0 B) Simple RLC Circuit with a 9V Battery: Lq′′ + Rq′ +(1/c)q = 9 7) (8 pts) Find all critical points for the given DE, draw a phase line for the system, then state the stability of each critical point. Logistic Equation: y′ = ry(1 −...
Solve the following differential equations 1. cos(t)y' - sin(t)y = t^2 2. y' - 2ty =...
Solve the following differential equations 1. cos(t)y' - sin(t)y = t^2 2. y' - 2ty = t Solve the ODE 3. ty' - y = t^3 e^(3t), for t > 0 Compare the number of solutions of the following three initial value problems for the previous ODE 4. (i) y(1) = 1 (ii) y(0) = 1 (iii) y(0) = 0 Solve the IVP, and find the interval of validity of the solution 5. y' + (cot x)y = 5e^(cos x),...
9) Find the maximal interval of existence, I, for each IVP given. A) (t^2 − 9)...
9) Find the maximal interval of existence, I, for each IVP given. A) (t^2 − 9) y′ − 7t^3 =√t, y(−2) = 12 B) sin(t) y′′ + ty′ − 18y = 1, y(4) = 9, y′(4) = −13
DIFFERENTIAL EQUATIONS 1. A force of 400 newtons stretches a spring 2 meters. A mass of...
DIFFERENTIAL EQUATIONS 1. A force of 400 newtons stretches a spring 2 meters. A mass of 50 kilograms is attached to the end of the spring and is initially released from the equilibrium position with an upward velocity of 10 m/s. Find the equation of motion. 2. A 4-foot spring measures 8 feet long after a mass weighing 8 pounds is attached to it. The medium through which the mass moves offers a damping force numerically equal to times the...
A mass of 1 slug, when attached to a spring, stretches it 2 feet and then...
A mass of 1 slug, when attached to a spring, stretches it 2 feet and then comes to rest in the equilibrium position. Starting at t = 0, an external force equal to f(t) = 4 sin(4t) is applied to the system. Find the equation of motion if the surrounding medium offers a damping force that is numerically equal to 8 times the instantaneous velocity. (Use g = 32 ft/s2 for the acceleration due to gravity.) What is x(t) ?...
Important Instructions: (1) λ is typed as lambda. (2) Use hyperbolic trig functions cosh(x) and sinh(x)...
Important Instructions: (1) λ is typed as lambda. (2) Use hyperbolic trig functions cosh(x) and sinh(x) instead of ex and e−x. (3) Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be Acos(x)+Bsin(x). (4) For polynomials use arbitrary constants in alphabetical order starting with highest power of x, for example, Ax2+Bx. (5) Write differential equations with leading term positive, so X′′−2X=0 rather than −X′′+2X=0. (6) Finally you need to simplify arbitrary constants. For...
1-Consider the following. 36y'' − y = 0, y(−4) = 1,   y'(−4) = −1 Find the...
1-Consider the following. 36y'' − y = 0, y(−4) = 1,   y'(−4) = −1 Find the solution of the given initial value problem. y(t) = ? 2- Consider the vibrating system described by the initial value problem. (A computer algebra system is recommended.) u'' + u = 9 cos ωt, u(0) = 5, u'(0) = 4 3-A spring is stretched 6 in. by a mass that weighs 8 lb. The mass is attached to a dashpot mechanism that has a...
Important Instructions: (1) λ is typed as lambda. (2) Use hyperbolic trig functions cosh(x) and sinh(x)...
Important Instructions: (1) λ is typed as lambda. (2) Use hyperbolic trig functions cosh(x) and sinh(x) instead of ex and e−x. (3) Write the functions alphabetically, so that if the solutions involve cos and sin, your answer would be Acos(x)+Bsin(x). (4) For polynomials use arbitrary constants in alphabetical order starting with highest power of x, for example, Ax2+Bx. (5) Write differential equations with leading term positive, so X′′−2X=0 rather than −X′′+2X=0. (6) Finally you need to simplify arbitrary constants. For...
Two waves traveling in opposite directions on a stretched rope interfere to give the standing wave...
Two waves traveling in opposite directions on a stretched rope interfere to give the standing wave described by the following wave function: y(x,t) = 4 sin⁡(2πx) cos⁡(120πt), where, y is in centimetres, x is in meters, and t is in seconds. The rope is two meters long, L = 2 m, and is fixed at both ends. In terms of the oscillation period, T, at which of the following times would all elements on the string have a zero vertical...
1. (5pts.) Compute the derivative dy/dx for y = 7√ 9π + x ^5 /6 +...
1. (5pts.) Compute the derivative dy/dx for y = 7√ 9π + x ^5 /6 + 27e^x . 3. (5pts.) Write the equation of the tangent line to the graph of y = 3 + 8 ln x at the point where x = 1. 4. (5pts.) Determine the slope of the tangent line to the curve 2x^3 + y^3 + 2xy = 14 at the point (1, 2). 5. (5pts.) Compute the derivative dw/dz of the function w =...