Question

1.Show that cos 2t, sin 2t, and e^5t are linearly independent and form a fundamental set of solutions for the equation: y ′′′ − 5y ′′ + 4y ′ − 20y = 0

2.Find the general solution to the equation: y ′′′ − y ′′ − 4y ′ + 4y = 0

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 homogeneous solutions to an ODE are sin(2t) and cos(2t).
Suppose that the forcing function is 1.5 cos(2t) what is an
appropriate form of the general solution?
y(t)=Acos2t +Bsin2t + C t cos(2t+ᶲ)
, (b) y(t)=Acos2t +Bsin2t
+ C cos2t + Dsin2t
y(t)=Acos2t
+Bsin2t,
(d) y(t)=Acos2t +Bsin2t + C cos(2t+ᶲ)
What is the total number of linearly independent solutions that
the following ODE must have?
y" +5y'+6xy=sinx
Two (b) Four
(c) Three
(d) Five

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))

a) Determine: L{t^3 e^2t+e^2t sin( 5t)} and b) Find L^(-1)
{(3s+2)/(s^2+2s+10)}

Consider the equation y'' + 4y = 0.
a) Justify why the functions y1 = cos(4t) and y2 = sin(4t) do not
constitute a fundamental set of solutions of the above
equation.
b) Find y1, y2 that constitute a fundamental set of solutions,
justifying your answer.

Show that the given functions y1 and y2 are solutions to the DE.
Then show that y1 and y2 are linearly independent. write the
general solution. Impose the given ICs to find the particular
solution to the IVP.
y'' + 25y = 0; y1 = cos 5x; y2 = sin 5x; y(0) = -2; y'(0) =
3.

Show that the set {sin x,sin 2x,sin 3x} is linearly independent
over R.

X'= 6x-5y+e^5t
Y'= x+4y
Find the general solution using undetermined coefficients

You found a particular solution to y''+2y'+5y=e^(2t) by guessing
a function based on the form of the nonhomogeneous term. Use the
same approach to find particular solutions to the equations
below.
6. y''+y'+5y=5sin(3t) + e^(2t)

Differential equations
Given that x1(t) = cos t is a solution of (sin t)x′′ − 2(cos
t)x′ − (sin t)x = 0, find a second linearly independent solution of
this equation.

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