Question

Consider the BVP y′′=-cy, y(0)=0 where c>0, y(b)=0.

For each c, find b>0 such that the BVP has at least two different solutions y1(t) and y2(t). Exhibit the solutions.

Answer #1

Find the function y1(t) which is the solution of 4y″+32y′+64y=0
with initial conditions y1(0)=1,y′1(0)=0.
y1(t)=?
Find the function y2(t) which is the solution of 4y″+32y′+64y=0
with initial conditions y2(0)=0, y′2(0)=1.
y2(t)= ?
Find the Wronskian of these two solutions you have found:
W(t)=W(y1,y2).
W(t)=?

Consider the initial value problem, ay''+by'+cy=0, y(0)=d,
y'(0)=f where a,b,c,d and f are constants which one of the
following could be a solution to the initial value problem? Give
breif exlpanation to why the correct answer can be a solution, and
why the others can not possibly satisfy the equation.
a. sin(t)+e^t
b. cost+e^tsint
c. cost+1
d. e^tcost

Consider the second-order homogeneous linear equation
y''−6y'+9y=0.
(a) Use the substitution y=e^(rt) to attempt to find two
linearly independent solutions to the given equation.
(b) Explain why your work in (a) only results in one linearly
independent solution, y1(t).
(c) Verify by direct substitution that y2=te^(3t) is a solution
to y''−6y'+9y=0. Explain why this function is linearly independent
from y1 found in (a).
(d) State the general solution to the given equation

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 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.

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

This is about differential equations.
Consider the equation for a damped oscillator with no external
force. Then the equation becomes my’’ +by’ + ky = 0. Show that if
y(t) is a solution to this equation, then cy(t) is a solution for
any constant c. Also show that if y1(t) and y2(t) are solutions to
this equation then c1y1(t) + c2y2(t) is also a solution for any c1
and c2.

Consider the differential equation x2y''+xy'-y=0,
x>0.
a. Verify that y(x)=x is a solution.
b. Find a second linearly independent solution using the method
of reduction of order. [Please use y2(x) =
v(x)y1(x)]

Please show all steps, thanks!!
a) Solve the BVP: y" + 2y' + y = 0, y(0) = 1, y(1) =3
b) Prove the superposition principle: suppose that the functions
y1(x) and y2(x) satisfy the homogenous equation of order two: ay''
+ by' + cy = 0.
Show that the following combinations also satisfy it:
constant multiple m(x) = k*y1(x)
sum s(x) = y1(x) + y2(x)

3. Consider a second order linear homogeneous equation Ay'' +
By' + Cy = 0 Suppose that e^at, e^bt and e^ct are solutions (where
a, b, c are constants). A. Show that e^at + e^bt + e^ct is also a
solution. B. Show that two of the numbers among a, b, c are
equal.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 9 minutes ago

asked 9 minutes ago

asked 16 minutes ago

asked 33 minutes ago

asked 52 minutes ago

asked 57 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago