Chapter 3.1, Problem 14E

(a)

To determine

A member of the family that satisfies the boundary conditions $x^2{y}^{″}-5x{y}^{\prime }+8y=24$, if the two parameters $y\left(-1\right)=0,y\left(1\right)=4$ are a solution of the indicated differential equation $y=c_1{x}^2+c_2{x}^4+3$ on the interval $\left(-\infty ,\infty \right)$.

(b)

To determine

A member of the family that satisfies the boundary conditions $x^2{y}^{″}-5x{y}^{\prime }+8y=24$, if the two parameters $y\left(0\right)=1,y\left(1\right)=2$ are a solution of the indicated differential equation $y=c_1{x}^2+c_2{x}^4+3$ on the interval $\left(-\infty ,\infty \right)$.

(c)

To determine

A member of the family that satisfies the boundary conditions $x^2{y}^{″}-5x{y}^{\prime }+8y=24$, if the two parameters $y\left(0\right)=3,y\left(1\right)=0$ are a solution of the indicated differential equation $y=c_1{x}^2+c_2{x}^4+3$ on the interval $\left(-\infty ,\infty \right)$.

(d)

To determine

A member of the family that satisfies the boundary conditions $x^2{y}^{″}-5x{y}^{\prime }+8y=24$, if the two parameters $y\left(1\right)=3,y\left(2\right)=15$ are a solution of the indicated differential equation $y=c_1{x}^2+c_2{x}^4+3$ on the interval $\left(-\infty ,\infty \right)$.