The given two-parameter family is a solution of the indicated differential equation on the interval (−∞, ∞). determine whether a member of the family can be found that satisfies the boundary conditions. y = c1x2 + c2x4 + 3; x2y'' − 5xy' + 8y = 24 (a) y(−1) = 0, y(1) = 8 a member can be found. a member cannot be found. (b) y(0) = 8, y(1) = 5 a member can be found. a member cannot be found. (c) y(0) = 3, y(1) = 0 a member can be found. a member cannot be found. (d) y(1) = 3, y(2) = 15 a member can be found. a member cannot be found.

We're told that the ODE

has solution

a. If and , then

but there is no solution to this system, so a member cannot be found.

b. If and , then

but the first equation is obviously false, so a member cannot be found.

c. If and , then

which has infinitely many solutions for , so a member can be found.

d. If and , then

so a member can be found.

answer: domain and range mean all possible values of x and y that could be substituted to the inequality. a solution means all possible values that make the inequality statement true. so while the range is still all real numbers, the solution is the shaded area, for which y will always be greater than the line of the function.