Use ‘Reduction of Order’ to find a second solution y2 to the
given ODEs:
(a) y′′+2y′+y=0, y1 =xe−x
(b) y′′+9y=0, y1 =sin3x
(c) x2y′′+2xy′−6y=0, y1 =x2
(d) xy′′ +y′ =0, y1 =lnx
Using the appropriate general solution you found in the problem
above, solve the following initial value problems. Sketch a graph
of the solution that captures the initial condition, the limiting
behaviour of the solution as t → ∞ and t → −∞ and the sign of the
solution (positive or negative) in these limits (look at the
dominant terms).
(c) 2y'' - 3y' + y = 0 , y(2) = 1 , y'(2) = 1
(e) 6y'' − 5y' +...
Find the closed formula solution to each of the following
recurrence relations with the given initial conditions. Use an
iterative approach and show your work! What is a_100? a)
a_n=a_(n-1)+2,a_0=3 b) a_n=a_(n-1)+2n+3,a_0=4 c)
a_n=2a_(n-1)-1,a_0=1 d) a_n=-a_(n-1),a_0=5
Find the closed formula solution to each of the following
recurrence relations with the given initial conditions. Use an
iterative approach and show your work! What is
a100 ?
an=an-1+2,
a0=3
an=an-1+2n+3,
a0=4
an=2an-1-1,
a0=1
an=-an-1,
a0=5
Use the one solution given below to find the general solution of
the differential equation below by reduction of order method:
(1 - 2x) y'' + 2y' + (2x - 3) y = 0
One solution: y1 = ex
Find the general solution of the following
differential equations (complementary function
+ particular solution). Find the particular solution by inspection
or by (6.18), (6.23),
or (6.24). Also find a computer solution and reconcile differences
if necessary, noticing
especially whether the particular solution is in simplest form [see
(6.26) and the discussion
after (6.15)].
(D2+2D+17)y = 60e−4x sin 5x