Let Y1(x) Y2(x) be two linearly independent solutions of the homogeneous equation y''+p(x)y'+q(x)y=0. Select all of the correct statements for the two-point boundary value problem

y''+p(x)y'+q(x)y=g(x)

y(0)=y0,

y'(L)=yL'

When this two-point boundary value problem has no solution, the corresponding homogeneous equation also has no solution.

When the matrix A =

Y1(0) Y2(0)
Y1(L) Y2(L)
is invertible, the nonhomogeneous two point value problem has a unique solution for any choices of g(x), y0, yL'

When this two-point boundary value problem has infinite many solutions, the corresponding homogeneous equation also has infinite many solutions.

This two-point boundary value problem could have no solution when the corresponding homogeneous equation has non trivial solutions.

The corresponding homogeneous equation is obtained by setting g(x)=0, y0=0, yL'=0

the system of inequity is

step-by-step explanation:

from the given graph it is noticed that we have two related lines. one is solid and another is dashed.

if a line passing through two points, then the equation of line is

the solid line passing through the points (0,4) and (4,0).

the related equation of solid line is .

similarly, find the related equation of dashed line. the dashed liune passing through the points (0,0) and (3,1).

the related equation of dashed line is

from the given graph it is noticed that the point (0,1) lies on the shaded region, therefore the point (0,1) must be satisfied by the related equations.

check the related equations by the point (0,1).

this statement is true if the sign of inequality is < instead of equality. since it is an solid line therefore the sign of inequality will be . the required inequality is

similarly, check the second related equation.

it is a dashed line, so this statement is true if the sign of inequality is > . the required inequality is

-8/2y-8=5/y+4-7y+8/y^2-16

we move all terms to the left:

-8/2y-8-(5/y+4-7y+8/y^2-16)=0

we can not solve this equation

1/2(2*4)=6 $$- agahajakanajak