Question

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 (−∞,∞)

Answer #1

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

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))

Verify that the given two-parameter family of functions is the
general solution of the non homogeneous differential equation on
the indicated interval.
y'' + y = sec x
y = c1cosx + c2sinx + xsinx + cosxln(cos x) ; (−π/2, π/2)

1) The given family of functions is the general solution of the
differential equation on the indicated interval. Find a member of
the family that is a solution of the initial-value problem.
y = c1 + c2 cos(x) + c3 sin(x),
(−∞, ∞);
y''' + y' = 0, y(π) =
0, y'(π) = 8, y''(π)
= −1
y =
2) Two chemicals A and B are combined to form
a chemical C. The rate, or velocity, of the reaction is
proportional to the...

Verify that the functions y1 = cos x − cos 2x and y2 = sin x −
cos 2x both satisfy the differential equation y′′ + y = 3 cos
2x.

<question 1>
find a solution for the given differential equation. if
possible, find a particular solution byspecifying the integral
coefficient. please show details of your work.
1) y'+(x^2)y=(e^(-x^3)sinhx)/3y^3
(hint : integral of sinh is cosh : you can prove this by using
the definition of cosh and sinh)
2) y''+2y'+y=2xsinx
3) y''+10y'+25y=100sinh(5x)
(hint : sin(5x)=1/2(e^5x-e^(-5x))

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.

y1 = 2 cos(x) − 1 is a particular solution for y'' + 4y = 6
cos(x) − 4. y2 = sin(x) is a particular solution for y''+4y = 3
sin(x). Using the two particular solutions, find a particular
solution for y''+4y = 2 cos(x)+sin(x)− 4/3 . Verify if the
particular solution satisfies the given DE.
[Hint: Rewrite the right hand of this equation in terms of the
given particular solutions to get the particular solution] Verify
if the particular...

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....

Find the general solution to the differential equation: y’’ – 6
y’ + 13y = 0
Find the general solution to the differential equation: y’’ +
5y’ + 4y = x + cos(x)

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