Question

Problem 8 Let a(t) =/= b(t) be given. The factorization for a second order ODE is commutative if (D + a(t) I) (D + b(t) I) y = (D + b(t) I) (D + a(t) I) y.

• Find condition on a(t) and b(t) so that the factorization is commutative.

• Find the fundamental set of solutions for a second order ODE that has a commutative factorization.

• Use the above results to find the fundamental set of solutions of y'' + 4 y' + 4 y = 0.

Answer #1

All the three parts are done separately

This is for Differential Equations. I was sick last week and
missed class so I do not understand the process behind solving this
question.
Problem 3 Consider the ODE t 2 y'' + 3 t y' + y =
0.
• Factorize the left hand side of the ODE. Hint: One of the
factors is (D + 2 t -1 I).
• Find the fundamental set of solutions.
• Solve the initial value problem t 2 y'' + 3 t...

Q.3 (Applications of Linear Second Order ODE): Consider the
‘equation of motion’ given by ODE d2x + ω2x = F0 cos(γt)
dt2
where F0 and ω ̸= γ are constants. Without worrying about those
constants, answer the questions (a)–(b).
(a) Show that the general solution of the given ODE is [2 pts]
x(t) := xc + xp = c1 cos(ωt) + c2 sin(ωt) + (F0 / ω2 − γ2)
cos(γt).
(b) Find the values of c1 and c2 if the...

solve the second order ode ( I have problem to choose right yp
for term has cosh or sinh)
y"-6y'+y=6 cosh x
y(0)=0.2
y'(0)=0.05

Suppose y(t) is governed by ODE t3y'''(t) +
6t2y''(t) + 4ty'(t) = 0. Solve this ODE by performing
the following procedure: 1) Let x = ln(t) and set y(t) = φ(x) =
φ(ln(t)), 2) Use chain rule of diﬀerentiation to calculate
derivatives y in terms derivatives of φ, 3) by appropriate
substitution, construct an ODE governing φ(x) and solve it, 4) use
this φ(x) to get back y(t).

($4.2 Reduction of Order):
(a) Let y1(x) = x be a solution of the homogeneous ODE xy′′
−(x+2)y′ + ((x+2)/x)y = 0. Use the reduction
of order to find a second solution y2(x), and write the general
solution.

In this problem, y = c1ex +
c2e−x is a two-parameter family of
solutions of the second-order DE y'' − y = 0.
Find a solution of the second-order IVP consisting of this
differential equation and the given initial conditions.
y(−1) = 8, y'(−1) = −8
y=

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 equation Ax = b by using the LU factorization given
for A.
A= [4 -6
4
= [1 0 0 [4 -6
4 [0
-12 15
-7
-3 1 0 0 -3
5 b= 12
12 -15
8]
3 -1 1] 0 0
1] -12]
Let Ly=b and Ux=y. Solve for x and y.
y=
x=

Given that y1 = t, y2 = t 2 are solutions to the homogeneous
version of the nonhomogeneous DE below, verify that they form a
fundamental set of solutions. Then, use variation of parameters to
find the general solution y(t).
(t^2)y'' - 2ty' + 2y = 4t^2 t > 0

.1.) Modelling using second order differential equations
a) Find the ODE that models of the motion of the dumped spring
mass system with mass m=1, damping coefficient c=3, and spring
constant k=25/4 under the influence of an external force F(t) = cos
(2t).
b) Find the solution of the initial value problem with x(0)=6,
x'(0)=0.
c) Sketch the graph of the long term displacement of the mass
m.

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