Question

In this problem, you will solve the following first order linear
ODE: y' + (1/x)y = (2/x^{2} )+ 1 with y(1) = 1.

a) Solve the complimentary equation

b) Use the solution to the complimentary equation to find the general solution

c) Use the initial conditions to find the specific solution

Answer #1

Consider the linear first order system [16]
x′ = x + y (1) y′ =4x−2y. (2)
(a) Determine the equilibria of System (1)-(2) as well as their
stability. [6]
(b) Compute the general solution of System (1)-(2). [6]
(c) Determine the solution of the initial value problem
associated with System (1)-(2), with initial condition x(0) = 1,
y(0) = 2.

Solve the first order ODE via series
y' - y = (e^x)

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...

10.16: Write a user-defined MATLAB function that solves a
first-order ODE by applying the midpoint method (use the form of
second-order Runge-Kutta method, Eqs(10.65),(10.66)). For function
name and arguments use [x,y]=odeMIDPOINT(ODE,a,b,h,yINI). The input
argument ODE is a name for the function that calculates dy/dx. It
is a dummy name for the function that is imported into odeMIDPOINT.
The arguments a and b define the domain of the solution, h is step
size; yINI is initial value. The output arguments, x...

Solve the 1st-order linear differential equation using an
integrating fac-
tor. For problem solve the initial value problem. For each
problem, specify the solution
interval.
dy/dx−2xy=x, y(0) = 1

5.1 Application of Linear Second Order ODE): Consider the
‘spring-mass system’ represented by an ODE x′′ (t) + 16x(t) = 5 sin
4t with ICs: x(0) = 2, x′ (0) = 1. Answer the questions
(a)–(c):
(a) Is there damping in the system? Why or why not?
(b) Is there resonance in the system? Why or why not?
(c) Solve the ODE.

($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.

Identify the type of ODE below (ex. Separable, Linear, Exact,
etc...) and then solve the initial value problem using the
appropriate technique (give an explicit final answer in the form
"y=...")
(x2+1)(dy/dx) + 8xy = -5x, y(0) = 10

1) Basic Euler’s Method:
y'+xysin/y+1 y(0)=1
a) What is the initial condition?
b) What order is this differential equation?
c) Is this an autonomous differential equation?
d) Is this a separable differential equation?
e) Find the general solution to the given differential equation,
by hand. You will not be able to completely solve for y(x) – that’s
ok. Write out all your work and attach it to your Questions
tab.
f) Using the initial condition, solve the initial value problem...

find a basis of solutions for the 12th order homogeneous linear ODE
with the following characteristic equation:
x^2(x+6)(x-1)^3(x^2-10x+41)(x^2+9)^2=0

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