Given an IVP an(x)dnydxn+an−1(x)dn−1ydxn−1+…+a1( x)dydx+a0(x)y=g(x)an(x)dnydxn+an−1( x)dn−1ydxn−1+…+a1(x)dydx+a0(x)y=g(x ) y(x0)=y0, y′(x0)=y1, ⋯, y(n−1)(x0)=yn−1y(x0)=y0, y′(x0)=y1, ⋯, y(n−1)(x0)=yn−1 If the coefficients an(x),…,a0(x)an(x),…,a0(x) and the right hand side of the equation g(x)g(x) are continuous on an interval II and if an(x)≠0an(x)≠0 on II then the IVP has a unique solution for the point x0∈Ix0∈I that exists on the whole interval II. Consider the IVP on the whole real line sin(x)d2ydx2+cos(x)dydx+sin(x)y=tan (x)sin⁡(x)d2ydx2+cos⁡(x)dydx+sin⁡(x )y=tan⁡(x) y(1)=11, y′(1)=4,y(1)=11, y′(1)=4, The Fundamental Existence Theorem for Linear Differential Equations guarantees the existence of a unique solution on the interval

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!

b if for most of the graph

d if y intercept

step-by-step explanation: