$(15$ points$).$ Consider the differential equation: y$^(\text{'}\text{'})+49$y$=0.$ a$.$ Find the solution to the differential equation using a method learned previously. b$.$ Use the facts that cosx$=\backslash$sum$\_($n$=0)^(\backslash$infty $)((-1)^($n$))/((2$n$)!)$x$^(2$n$)$ and sinx$=\backslash$sum$\_($n$=0)^(\backslash$infty $)((-1)^($n$))/((2$n$+1)!)$x$^(2$n$+1)$ and what you learned about power series in Calculus II to write your answer to part a in terms of power series. c$.$ Use the methods covered in Chapter $5$ to find the power series solution about the ordinary point x$=0.$ Write at least four non$-$zero terms of each solution. You do NOT need to write a general summation. d$.$ CLEARLY DEMONSTRAT$($E$)/($E$)$XPLAIN HOW THE SOLUTION FOUND IN PART A AND REPRESENTED IN TERMS OF POWER SERIES IN PART B IS EQUIVALENT TO THE POWER SERIES SOLUTION IN PART C$.$

$(10$ points$).$ For the following differential equation, find two linearly independent power series solutions about x$=0.$ Write at least $4$ non$-$zero terms of each solution. If it is not possible to find $4$ non$-$zero terms, write as many as you can find. $($x$^(2)-5)$y$^(\text{'}\text{'})-4$xy$^(\text{'})-2$y$=0$