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I am a PhD student in the department of Mathematics at the University of Georgia. My current research interests involve analytic number theory, particularly statistics of arithmetic functions. My past research endeavors involve modeling of infectious disease epidemiology, analytic combinatorics, and combinatorial optimization. When I am not teaching or working on my degree I enjoy running, picking the guitar, playing bridge, and spending time with my beautiful wife and daughter.

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length $ a $ if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse.

$$\begin{array}{l}{y^{\prime \prime}(t)=\cos (t-y)+y^{2}(t)} \\ {y(0)=1, \quad y^{\prime}(0)=0}\end{array}$$

$$\begin{array}{l}{y^{(6)}(t)=\left[y^{\prime}(t)\right]^{3}-\sin (y(t))+e^{2 t}} \\ {y(0)=y^{\prime}(0)=\cdots=y^{(5)}(0)=0}\end{array}$$

$$\begin{array}{ll}{3 x^{n}+5 x-2 y=0 ;} & {x(0)=-1, \quad x^{\prime}(0)=0} \\ {4 y^{\prime \prime}+2 y-6 x=0 ;} & {y(0)=1, \quad y^{\prime}(0)=2}\end{array}$$

Sturm Liouville Form. A second-order equation is said to be in Sturm Liouville form if it is expressed as$$\left[p(t) y^{\prime}(t)\right]^{\prime}+q(t) y(t)=0$$

SturmLiouville Form. A second-order equation is said to be in SturmLiouville form if it is expressed as$$\left[p(t) y^{\prime}(t)\right]^{\prime}+q(t) y(t)=0$$Show that the substitutions $x_{1}=y, x_{2}=p y^{\prime}$ result inthe normal form$$\begin{aligned} x_{1}^{\prime} &=x_{2} / p \\ x_{2}^{\prime} &=-q x_{1} \end{aligned}$$