Question

Find a linear differential operator that annihilates the given function. (Use D as the differential operator)

x cos(2x)

Verify that the differential operator you obtained annihilates the given function.

Answer #1

Verify that the given functions form a fundamental set of
solutions of the differential equation on the indicated interval.
Form the general solution.
1.) y'' − 4y = 0; cosh 2x, sinh 2x, (−∞,∞)
2.) y^(4) + y'' = 0; 1, x, cos x, sin x (−∞,∞)

The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order or formula (5)
in Section 4.2,
y2 = y1(x)
e−∫P(x) dx
y
2
1
(x)
dx (5) as instructed, to find a
second solution y2(x).
y'' + 36y = 0; y1 =
cos(6x)
y2 =
2) The indicated function y1(x) is a solution of the
given differential equation. Use reduction of order or formula (5)
in Section 4.2,
y2 = y1(x)
e−∫P(x) dx
y
2
1...

Use the Integrating Factor Technique to find the solution to the
first-order linear differential equation
(1 − ? ∙ ?in(?)) ∙ ?? − cos(?) ∙ ?? = 0 with ?(?) = 1

Find a particular solution of the given differential equation.
Use a CAS as an aid in carrying out differentiations,
simplifications, and algebra. y(4) + 2y'' + y = 8 cos(x) − 12x
sin(x)

The indicated function y1(x) is a solution of the given
differential equation.
Use reduction of order or formula (5) in Section 4.2,
y2 = y1(x) e−∫P(x) dx y 2 1 (x) dx (5)
as instructed, to find a second solution y2(x). y'' + 100y = 0;
y1 = cos 10x
I've gotten to the point all the way to where y2 = u y1, but my
integral is wrong for some reason
This was my answer
y2= c1((sin(20x)+20x)cos10x)/40 + c2(cos(10x))

DIFFERENTIAL EQUATIONS INDETERMINATE COEFFICIENTS given the
function g (x) determine the proposed particular solution Yp if its
possible
g(x)= -6xe^(2x)
g(x)= e^(-x) cos(2x)
g(x)= 9x^(2) -17xe^(x/2) sin(x)
g(x)= 5x^(-2) +8x^(-1) -(7/3) + 3 sqr(2)x +6x^(2)

Find an appropriate integrating factor that will convert the
given not exact differential equation cos x d x + ( 1 + 2 y ) sin
x d y = 0 into an exact one. Then solve the new exact
differential equation.

Find the solution to the linear system of differential
equations
{x′ = 6x + 4y
{y′=−2x
satisfying the initial conditions x(0)=−5 and
y(0)=−4.
x(t) = _____
y(t) = _____

Solve the given differential equation by undetermined
coefficients.
y'' + 2y' + y = 2 cos x − 2x sin x

Verify that the given function is a solution and use Reduction
of Order to find a second linearly independent solution.
a. x2y′′ −2xy′ −4y = 0, y1(x) = x4.
b. xy′′ − y′ + 4x3y = 0, y1(x) =
sin(x2).

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