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**Research sample 1**

This are equations involving one or more derivatives.

For instance; y^{//}+q (t) y=0s

Research in such kind of equations should include the following:

- A statement of fact should be included
- Its proof should be there too

This type of equations come into two flavors. Namely;

- The first one is the ordinary while the second is the partial. Ordinary equations are functions of one variable while the partial has two or more variables.
^{1}According to Lodes, the classical form of this equations is:

P_{n}y^{ (n)} +p_{n-1}y^{ (n-1)} +…p1y^{’}+p0y=f

The functions in this case p^{’}s, f and y^{’}s are the functions. N is a positive integer while y^{ (j)} is the j-th derivative of the y. for us to determine the properties of solutions y, then the coefficients p_{j}, f have to be used.^{1} In this case, any question to do with p_{j} and f can comfortably be answered.

Elementary function can be differentiated through;

The power rule

Derivative of x is given by:

d/dx*x^{p}=px^{p-1}

Trigonometric functions

The derivatives of sin x and cos x are:

(sinx)^{’}=cos x, (cos x)^{’}=-sin x

Exponential

Derivatives of e^{x} and ln x are (e^{x})^{’}, (ln x)^{’}=1/x

**Research sample 2 **

Differential equations can be applied to a variety of topics ranging from physics to population growth to stock market. They predict future outcomes such as disease spread or changes in populations over time.^{2}

Rules applied are;

The power rule

Derivative of x is given by:

d/dx*xp=pxp-1

Trigonometric functions

The derivatives of sin x and cos x are:

(sinx)’=cos x, (cos x)’=-sin x

Exponential

Derivatives of ex and ln x are (ex)’, (ln x)’=1/x

Generally a differential is simply an equation involving unknown functions and its derivatives. Mathematically they are equations for an unknown functions of one or several variables that relates the values of the function itself and its derivatives of various orders.^{2} Differential equations fall under two categories that is ordinary differential equations (ODE) and partial differential equations (PDE) and their distinctions being that ODEs involve unknown functions of one independent variable while in PDEs involve unknown functions of more than one independent variable.

An(x) yn+an-1+…+a1(x) y’+a0(x) y=Q(x) [1]

When the derivatives of y are evaluated at a single point an nth order ODE will have

Y(x0), y’(x0)…, y (n-1)

A good example to represent BVP equation is

Y’’+p(x) y’+q(x) y=g(x)

The conditions are strictly;

Y (k) =k, y(C) =c

Bibliography

- Fenichel, Neil. “
*Geometric singular perturbation theory for ordinary differential “*Journal of Differential Equations 31, no. 1 (1979): 53-98. - Kwong, Man Kam, and Anton Zettl.
*Norm inequalities for derivatives and*No. CONF-8707135-1. Northern Illinois Univ., Dekalb (USA). Dept. of Mathematical Sciences; Argonne National Lab., IL (USA), 1987.