Hw$21-6\mathrm{.}1-$BVP$-$SEP: Problem $1$

Problem Value: $10$ point$($s$).$ Problem Score: $0\%.$ Attempts Remaining: $20$ attempts.

Help Entering Answers See Examples $6\mathrm{.}1\mathrm{.}2-6\mathrm{.}1\mathrm{.}3,$ in Section $6\mathrm{.}1,$ in the MTH $235$ Lecture Notes.

$(10$ points$)$

Consider the BVP for the function $y$ given by

$y^{\text{'}\text{'}}+49y=0,y(0)=4,y(\frac{3}{7})=2$

$($a$)$ Find $r_1,r_2,$ roots of the characteristic polynomial of the equation above.

$r_1,r_2$

$$

$($b$)$ Find a set of real$-$valued fundamental solutions to the differential equation above.

$y_1(x)=$

$y_2(x)=$

$($c$)$ Find all solutions $y$ of the boundary value problem.

$y(x)=$

$$

Note $1$: If there are no solutions, type No Solution.

Note $2$: If there are infinitely many solutions, use $k$ for the arbitrary constant.