In: Advanced Math

The *Bessel equation* of order *p* is
t^{2}y" + ty' + (t^{2} - p^{2})y = 0. In
this problem, assume that p = 1/2:

a.) Show that y_{1} = sin(t / sqrt(t)) and y_{2}
= cos(t / sqrt(t)) are linearly independent solutions for 0 < t
< infinity.

b.) Use the result from part (a), and the preamble in Exercise
3, to find the general solution of t^{2}y" + ty' +
(t^{2} - (1/4))y = t^{3/2}cos(t). (answer should
be: 1/2 sin(t) sqrt(t))

**Preamble of Exercise 3: The formula for a particular solution
given in (3.42) applies to the more general problem of solving y" +
p(t)y' + q(t)y = f(t). In this case, y_{1} and
y_{2} are independent solutions of the associated
homogeneous equation y" + p(t)y' + q(t)y = 0.

Please show work!

for y(t) function ty'' - ty' + ty = 0, y(0)= 0 , y'(0)= 1
solve this initial value problem by using Laplace Transform.
(The equation could have been given such as "y'' - y' + y = 0" but
it is not. Please, be careful and solve this question step by
step.) )

Solve the following differential equation, using laplace
transforms:
y''+ty-y=0
where
y(0)=0
and
y'(0)=3

a) Verify that the indicial equation of Bessel's equation of
order p is (r-p)(r+p)=0
b) Suppose that p is not an integer. Carry out the computation
to obtain the solutions y1 and y2 above.

Consider the homogeneous second order equation
y′′+p(x)y′+q(x)y=0. Using the Wronskian, find functions p(x) and
q(x) such that the differential equation has solutions sinx and
1+cosx. Finally, find a homogeneous third order differential
equation with constant coefficients where sinx and 1+cosx are
solutions.

Consider the following second-order ODE,
y"+1/4y=0
with y(0) = 1 and y'(0)=0.
Transform this unique equation into a system of two 1st-order
ODEs.
Solve the obtained system for t in [0,0.6] with h = 0.2 by
MATLAB using Euler Method or Improved Euler Method.

Consider the linear transformation T : P2 ? P2 given by T(p(x))
= p(0) + p(1) + p 0 (x) + 3x 2p 00(x). Let B be the basis {1, x,
x2} for P2.
(a) Find the matrix A for T with respect to the basis B.
(b) Find the eigenvalues of A, and a basis for R 3 consisting of
eigenvectors of A.
(c) Find a basis for P2 consisting of eigenvectors for T.

Use method of Frobenius to find one solution of Bessel's
equation of order p: x^2y^''+xy^'+(x^2-p^2)y=0

3. Consider4 the homogenous linear second order differential
equation
y′′ − 2y′ + y = 0 (⋆)
(a) Verify that the function y = e^x is a solution of equation
(⋆) on the interval (−∞, ∞).
(b) Verify that the function y = xex is a solution of equation
(⋆) on the interval (−∞, ∞).
(c) Verify that y = 7e^x + (5xe)^x is a solution of equation
(⋆) on the interval (−∞, ∞).
(d) Assume that c and d...

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2
y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t
y′ = 5t4y, y(0) = 1, solution: y(t) = et5
y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3
For the IVPs above, make a log-log plot of the error of Backward
Euler and Implicit Trapezoidal Method, at t = 1 as a function of
hwithh=0.1×2−k for0≤k≤5.

y′ = t, y(0) = 1, solution: y(t) = 1+t2/2
y′ = 2(t + 1)y, y(0) = 1, solution: y(t) = et2+2t
y′ = 5t4y, y(0) = 1, solution: y(t) = et5
y′ = t3/y2, y(0) = 1, solution: y(t) = (3t4/4 + 1)1/3
For the IVPs above, make a log-log plot of the error of Explicit
Trapezoidal Rule at t = 1 as a function ofhwithh=0.1×2−k
for0≤k≤5.

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