Question

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,∞) ?

Answer #1

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.

Two solutions to the diﬀerential equation y00 + 2y0 + y = 0 are
y1(t) = e−t and y2(t) = te−t. Verify that y1(t) is a solution and
show that y1,y2 form a fundamental set of solutions by computing
the Wronskian

given that y1=xcos(lnx)and y2=xsin(lnx)form a fundamental set of
solutions to x^2y''-xy'+2y=0,find general solution to
x^2y''-xy'+2y=xlnx

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:
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 = ?????

Consider the differential equation t 2 y" + 3ty' + y = 0, t >
0. (a) Check that y1(t) = t −1 is a solution to this equation. (b)
Find another solution y2(t) such that y1(t) and y2(t) are linearly
independent (that is, y1(t) and y2(t) form a fundamental set of
solutions for the differential equation)

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....

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

Differential Equations problem
If y1= e^-x is a solution of the differential equation
y'''-y''+2y=0 . What is the general solution of the differential
equation?

B. a non-homogeneous differential equation, a complementary
solution, and a particular solution are given. Find a solution
satisfying the given initial conditions.
y''-2y'-3y=6 y(0)=3 y'(0) = 11 yc=
C1e-x+C2e3x
yp = -2
C. a third-order homogeneous linear equation and three linearly
independent solutions are given. Find a particular solution
satisfying the given initial conditions
y'''+2y''-y'-2y=0, y(0) =1, y'(0) = 2, y''(0) = 0
y1=ex, y2=e-x,,
y3= e-2x

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