Question

Consider the differential equation: 66t^2y''+12t(t-11)y'-12(t-11)y=5t^3, . You can verify that y1 = 5t and y2 = 4te^(-2t/11)satisfy the corresponding homogeneous equation.

The Wronskian W between y1 and y2 is W(t) = (-40/11)t^2e^((-2t)/11)

Apply variation of parameters to find a particular solution.

yp = ?????

Answer #1

Given y1(t)=t^2 and y2(t)=t^-1 satisfy the corresponding
homogeneous equation of
t^2y''−2y=2−t3, t>0
Then the general solution to the non-homogeneous equation can be
written as y(t)=c1y1(t)+c2y2(t)+yp(t)
yp(t) =

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.

Let y1 and y2 be two solutions of the equation y'' + a(t)y' +
b(t)y = 0 and let W(t) = W(y1, y2)(t) be the Wronskian. Determine
an expression for the derivative of the Wronskian with respect to t
as a function of the Wronskian itself.

Consider the differential equation
L[y] = y′′ + p(t)y′ + q(t)y = f(t) + g(t), and suppose L[yf] =
f(t) and L[yg] = g(t).
Explain why yp = yf + yg is a solution to L[y] = f + g.
Suppose y and y ̃ are both solutions to L[y] = f + g, and
suppose
{y1, y2} is a fundamental set of solutions to the homogeneous
equation L[y] = 0. Explain why
y = C1y1 + C2y2 + yf...

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

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

a) Find the general solution of the differential equation
y''-2y'+y=0
b) Use the method of variation of parameters to find the general
solution of the differential equation y''-2y'+y=2e^t/t^3

it can be shown that y1=x^(−2), y2=x^(−5) and y3=2
are solutions to the differential equation x^2D^3y+10xD^2y+18Dy=0
on (0,∞)
What does the Wronskian of y1,y2,y3 equal?
W(y1,y2,y3) =
Is {y1,y2,y3} a fundamental set for x^2D^3y+10xD^2y+18Dy=0 on
(0,∞) ?

The function y1(t) = t is a solution to the
equation.
t2 y'' + 2ty' - 2y = 0, t > 0
Find another particular solution y2 so that
y1 and y2 form a fundamental set of
solutions. This means that, after finding a solution y2,
you also need to verify that {y1, y2} is
really a fundamental set of solutions.

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)= ?

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 1 minute ago

asked 1 minute ago

asked 9 minutes ago

asked 10 minutes ago

asked 12 minutes ago

asked 13 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago