Question

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

Answer #1

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 = 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) 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
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 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) 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 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) 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 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 + 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 =...

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