A differential equation and a nontrivial solution f are given below. Find a second linearly independent solution using reduction of order. Assume that all constants of integration are zero.
tx" - (3t + 1)x' + 3x = 0; t>0, f(t) = 4e³ᵗ

The second linearly independent solution is

g(t) = = -(4/9)(3t + 1)

Step-by-step explanation:

Given the differential equation

tx'' - (3t + 1)x' + 3x = 0 (1)

and a solution

f(t) = 4e^(3t)

We want to find a second linearly independent solution g(t), using the method of reduction of order.

Let this second solution be

x = uf(t)

x = u. 4e^(3t) (2)

x' = u'. 4e^(3t) + u. 12e^(3t) (3)

x'' = u''. 4e^(3t) + u'. 12e^(3t) + u'. 12e^(3t) + u. 36e^(3t)

= 4u''e^(3t) + 24u'e^(3t) + 36ue^(3t) (4)

Using the values of x, x', and x'' in (2), (3), and (4) in (1), we have

t[4u''e^(3t) + 24u'e^(3t) + 36ue^(3t)] - (3x + 1)[u'. 4e^(3t) + u. 12e^(3t)] + 3u. 4e^(3t) = 0

4tu''e^(3t) + (12t - 4)u'e^(3t) = 0(5)

Let w = u'

then w' = u''

(5) now becomes

4tw'e^(3t) + (12t - 4)we^(3t) = 0

4tw'e^(3t) = (4 - 12t)we^(3t)

w'/w = (4 - 12t)/4t = (1/t) - 3

Integrating this, assuming all constants of integration are 0, we have

lnw = lnt - 3t

w = e^(lnt - 3t) = e^(lnt)e^(-3t)

w = te^(-3t)

But remember w = u'

=> u' = te^(-3t)

Integrating this, by part, taking constant of integration as 0, we have

u = -(1/9)(3t + 1)e^(-3t)

Using this in (2)

x = 4 [-(1/9)(3t + 1)e^(-3t)]e^(3t)

= -(4/9)(3t + 1)

And this is what we are looking for.

a.   we have two

lines:   y = 4-x   and  

y = 8-x^-1

given two simultaneous equations that are both to be

true, then the solution is the points where the lines cross. the intersection is

where the two equations are equal. therefore the solution that works for both

equations is when

4-x = 8-x^-1

this is where the two lines will cross and that is the

common point that satisfies both equations.

b.   4-x = 8-x^-1

  x           4-x       8-x^-1

-3           7        

8.33

-2           6         8.5

-1           5         9

  0           4        

-

  1           3        

7

  2           2        

7.5

  3           1         7.67

the table shows that none of the x values from -3 to 3 is

the solution because in no case does

4-x = 8-x^-1

to find the solution we need to rearrange the equation to

find for x:

4-x = 8-x^-1

multiply both sides with x:

4x-x^2 = 8x-1

x^2+4x-1=0

x= -4.236, 0.236

therefore there are two points that satisfies the

equation.

find y:  

x=-4.236

y = 4-x   = 4 – (-4.236)

= 8.236

y = 8-x^-1 =   .236)^-1

= 8.236

x=0.236

  y = 4-x   = 4 – (0.236) = 3.764

y = 8-x^-1 =   8-(0.236)^-1

= 3.764

thus the two lines cross at 2 points:

(-4.236, 8.236) & (0.236, 3.764)

c.   to solve

graphically the equation 4-x = 8-x^-1

we would graph both lines: y = 4-x   and   y

= 8-x^-1

the point on the graph where the lines cross is the

solution to the system of equations.

just graph the points on part b on a cartesian coordinate

system and extend the two lines.   the

solution is, as stated, the point where the two lines cross on the graph.

step-by-step explanation:

mark brainliest!

answer is 1

step-by-step explanation: