Question

The
following are solutions to homogeneous linear differential
equations. Obtain the corresponding differential equation with
real, constant cocfficients that is satisfied by the given
function:

a) y=4e^2x+3e^-x

b) y=x^2-5sin3x

c) y=-2x+1/2e^4x

d) y=xe^-xsin2x+3e^-xcos2x

Answer #1

Second-Order Linear Non-homogeneous with Constant Coefficients:
Find the general solution to the following differential equation,
using the Method of Undetermined Coefficients.
y''− 2y' + y = 4x + xe^x

Question 4
a) Solve y```` -4y```-5y``+36Y`− 36?=0 knowing that xe^2x it's a
solution
b) Determine the general solution for a linear homogeneous
differential equation in y(x) de fourth order, with real
coefficients, knowing that a solution is given by x^3e^4x.
Leave the calculations recorded in the
activity

a) The homogeneous and particular solutions of
the differential equation ay'' + by' + cy = f(x) are, respectively,
C1exp(x)+C2exp(-x) and 3x^3. Give the complete solution y(x) of the
differential equation.
b) If the force f(x) in the equation given in a)
is instead f(x) = f1(x) + f2(x) + f3(x), where f1(x), f2(x), and
f3(x) are generic forces, what would be the particular
solution?
c) The homogeneous solution of a forced oscillator
is cos(t) + sin(t), what is the...

Write down a homogeneous second-order linear differential
equation with constant coefficients whose solutions are:
a. e^-xcos(x) , e^-xsin(x)
b. x , e^x

1. If x1(t) and x2(t) are solutions to the differential
equation
x" + bx' + cx = 0
is x = x1 + x2 + c for a constant c always a solution? Is the
function y= t(x1) a solution?
Show the works
2. Write sown a homogeneous second-order linear differential
equation where the system displays a decaying oscillation.

B. a non-homogeneous differential equation, a complementary
solution, and a particular solution are given. Find a solution
satisfying the given initial conditions.
y''-2y'-3y=6 y(0)=3 y'(0) = 11 yc=
C1e-x+C2e3x
yp = -2
C. a third-order homogeneous linear equation and three linearly
independent solutions are given. Find a particular solution
satisfying the given initial conditions
y'''+2y''-y'-2y=0, y(0) =1, y'(0) = 2, y''(0) = 0
y1=ex, y2=e-x,,
y3= e-2x

) Check that each of the following functions solves the
corresponding differential equation, by computing both the
left-hand side and right-hand side of the differential
equation.
(a) y = cos2 (x) solves dy/dx = −2 sin(x) √y
(b) y = 4x + 1/x solves x dy dx + 2/x = y
(c) y = e x 2+3 solves dy/dx = 2xy
(d) y = ln(1 + x 2 ) solves e y dy dx = 2x

Consider the differential equation x^2y′′ − 3xy′ − 5y = 0. Note
that this is not a constant coefficient differential equation, but
it is linear. The theory of linear differential equations states
that the dimension of the space of all homogeneous solutions equals
the order of the differential equation, so that a fundamental
solution set for this equation should have two linearly fundamental
solutions.
• Assume that y = x^r is a solution. Find the resulting
characteristic equation for r....

Non homogeneous eq w constant; undetermined coefficients
Find the general solution:
1) y" + 4y' + 4y = xe^−x.
2) y" + 2y' + 5y = e^2x cos x.
Determine a suitable form for a particular solution z = z(x) of
the given equations
1) y" + 2y' = 2x + x^2e^−3x + sin 2x.
2) y" − 5y' + 6y = 2e^2x cos x − 3xe^3x + 5.
3) y" + 5y' + 6y = 2e^2x cos x −...

The indicated functions are known linearly independent solutions
of the associated homogeneous differential equation on (0, ∞). Find
the general solution of the given nonhomogeneous equation.
x2y'' + xy' + y = sec(ln(x))
y1 = cos(ln(x)), y2 = sin(ln(x))

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