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.

By checking all the options and finding number of solution we can say that differential equation [tex]$x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24$\\$\\[/tex] have a member of the family when y(1)=3 and y(2)=15

What is a differential equation?

An equation of function and their derivatives is called differential equation .

Given differential equation

[tex]$x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24$\\$\\[/tex]

And solution of this equation is

[tex]$y=C_{1} x^{2}+C_{2} x^{4}+3$[/tex]

(a)

[tex]y(-1)=0$ \ \ \text{and }\ $y(1)=8$[/tex]

Putting these values in solution

[tex]C_{1}+C_{2}+3 =0 \\ \text{and }C_{1}+C_{2}+3=8[/tex]

Both lines are parallel so there is no solution in this case . Hence a member can not be found

(b)

[tex]y(0)=8$\text{ and } $y(1)=5$[/tex]

Putting these values in solution

[tex]8=3 \\\text{and }C_{1}+C_{2}+3=5[/tex]

Which is not possible so there is no solution in this case .Hence a member can not be found

(c)

[tex]y(0)=3 \ \ \text{and } y(1)=0[/tex]

Putting these values in solution

[tex]3=3 \\C_{1}+C_{2}+3=0[/tex]

These equation have infinitely many solution .Hence infinite member can be found

(d)

[tex]y(1)=3$ \text{ and } $y(2)=15[/tex]

Putting these values in solution

[tex]C_{1}+C_{2}+3 =3\ \ \ \ \Rightarrow C_{1}+C_{2} =0 \\\\\4 C_{1}+256 C_{2}+3=15 \\\\\Rightarrow 4 C_{1}+256 C_{2}=12 \\\\\Rightarrow C_{1}+64 C_{2}=3 \\[/tex]

as these system of equation has unique solution .Hence one member can be found

By checking all the options and finding number of solution we can say that differential equation [tex]$x^{2} y^{\prime \prime}-5 x y^{\prime}+8 y=24$\\$\\[/tex] have a member of the family when y(1)=3 and y(2)=15 .

To learn more about differential equation visit :https://brainly.com/question/1164377