Question

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.

Answer 1

We're told that the ODE

$x^2y''-5xy'+8y=24$

has solution

$y=C_1x^2+C_2x^4+3$

a. If $y(-1)=0$ and $y(1)=8$, then

$\begin{cases}0=C_1+C_2+3\\8=C_1+C_2+3\end{cases}$

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

b. If $y(0)=8$ and $y(1)=5$, then

$\begin{cases}8=3\\5=C_1+C_2+3\end{cases}$

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

c. If $y(0)=3$ and $y(1)=0$, then

$\begin{cases}3=3\\0=C_1+C_2+3\end{cases}$

which has infinitely many solutions for $C_1,C_2$, so a member can be found.

d. If $y(1)=3$ and $y(2)=15$, then

$\begin{cases}3=C_1+C_2+3\\15=4C_1+256C_2+3\end{cases}\implies C_1=-\dfrac5{84},C_2=\dfrac5{84}$

so a member can be found.