Question

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.

Answer #1

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))

Given that y1 = t, y2 = t 2 are solutions to the homogeneous
version of the nonhomogeneous DE below, verify that they form a
fundamental set of solutions. Then, use variation of parameters to
find the general solution y(t).
(t^2)y'' - 2ty' + 2y = 4t^2 t > 0

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...

if y1 and y2 are linearly independent solutions of t^2y'' + 3y'
+ (2 + t)y = 0 and if W(y1,y2)(1)=3, find W(y1,y2)(3).
ROund your answer to the nearest decimal.

Use variation of parameters to find a general solution to the
differential equation given that the functions y1 and y2 are
linearly independent solutions to the corresponding homogeneous
equation for t>0.
y1=et y2=t+1
ty''-(t+1)y'+y=2t2

Use variation of parameters to find a general solution to the
differential equation given that the functions y 1 and y 2 are
linearly independent solutions to the corresponding homogeneous
equation for t>0.
ty"-(t+1)y'+y=30t^2 ; y1=e^t , y2=t+1
The general solution is y(t)= ?

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.

let
y1=e^x be a solution of the DE 2y''-5y'+3y=0 use the reduction of
order method to find a second linearly independent solution y2 of
the given DE

y = c1 cos(5x) + c2 sin(5x) is a two-parameter family of
solutions of the second-order DE y'' + 25y = 0. If possible, find a
solution of the differential equation that satisfies the given side
conditions. The conditions specified at two different points are
called boundary conditions. (If not possible, enter
IMPOSSIBLE.)
y(0) = 1, y'(π) = 7
y =

In this problem verify that the given functions y1 and y2
satisfy the corresponding homogeneous equation. Then find a
particular solution of the nonhomogeneous equation.
x^2y′′−3xy′+4y=31x^2lnx, x>0, y1(x)=x^2, y2(x)=x^2lnx. Enter an
exact answer.

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