Dada la ecuacion de la conica -6x_1^2 + 20 x_1 x_2 + 8x_2^2 + 30 x_1 - 16 x_2 = 15 encontrar, por transformacion de efes, primero en traslacion y luego en rotacion Centro de la coniea Orintacion de sus ejes principales Identifiear el tipo de conica: elipse, parabola o hyoerbola Solution Ans- the problem of a vibrating string such as that of a musical instrument was studied by Jean le Rond d\'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.[4][5][6][7] In 1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler discovered the three-dimensional wave equation.[8] The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection with their studies of the tautochrone problem. This is the problem of determining a curve on which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the starting point. Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed Lagrange\'s method and applied it to mechanics, which led to the formulation ofLagrangian mechanics. Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic Theory of Heat),[9] in which he based his reasoning on Newton\'s law of cooling, namely, that the flow of heat between two adjacent molecules is proportional to the extremely small difference of their temperatures. Contained in this book was Fourier\'s proposal of hisheat equation for conductive diffusion of heat. This partial differential equation is now taught to every student of mathematical physics. Example[edit] For example, in classical mechanics, the motion of a body is described by its position and velocity as the time value varies. Newton\'s laws allow (given the position, velocity, acceleration and various forces acting on the body) one to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly. An example of modelling a real world problem using differential equations is the determination of the velocity of a ball falling through the air, considering only gravity and air resistance. The ball\'s acceleration towards the ground is the acceleration due to gravity minus the acceleration due to air resistance. Gravity is considered constant, and air resistance may be modeled as proportional to the ball\'s velocity. This means that the ball\'s acceleration, which is a derivative of its velocity, depends on the velocity (and the velocity depends on time). Finding the velocity as a function of time involves solving a differential equation and verifying its validity. Types[edit] Differential equations can be divided into several types. Apart from describing the properties of the equation itself, these classes of differential equations can help inform the choice of approach to a solution. Commonly us.

1. Dada la ecuacion de la conica -6x_1^2 + 20 x_1 x_2 + 8x_2^2 + 30 x_1 - 16 x_2 = 15
encontrar, por transformacion de efes, primero en traslacion y luego en rotacion Centro de la
coniea Orintacion de sus ejes principales Identifiear el tipo de conica: elipse, parabola o
hyoerbola
Solution
Ans-
the problem of a vibrating string such as that of a musical instrument was studied by Jean le
Rond d'Alembert, Leonhard Euler, Daniel Bernoulli, and Joseph-Louis Lagrange.[4][5][6][7] In
1746, d’Alembert discovered the one-dimensional wave equation, and within ten years Euler
discovered the three-dimensional wave equation.[8]
The Euler–Lagrange equation was developed in the 1750s by Euler and Lagrange in connection
with their studies of the tautochrone problem. This is the problem of determining a curve on
which a weighted particle will fall to a fixed point in a fixed amount of time, independent of the
starting point.
Lagrange solved this problem in 1755 and sent the solution to Euler. Both further developed
Lagrange's method and applied it to mechanics, which led to the formulation ofLagrangian
mechanics.
Fourier published his work on heat flow in Théorie analytique de la chaleur (The Analytic
Theory of Heat),[9] in which he based his reasoning on Newton's law of cooling, namely, that
the flow of heat between two adjacent molecules is proportional to the extremely small
difference of their temperatures. Contained in this book was Fourier's proposal of hisheat
equation for conductive diffusion of heat. This partial differential equation is now taught to every
student of mathematical physics.
Example[edit]
For example, in classical mechanics, the motion of a body is described by its position and
velocity as the time value varies. Newton's laws allow (given the position, velocity, acceleration
and various forces acting on the body) one to express these variables dynamically as a
differential equation for the unknown position of the body as a function of time.
In some cases, this differential equation (called an equation of motion) may be solved explicitly.
An example of modelling a real world problem using differential equations is the determination
of the velocity of a ball falling through the air, considering only gravity and air resistance. The
ball's acceleration towards the ground is the acceleration due to gravity minus the acceleration
due to air resistance.
Gravity is considered constant, and air resistance may be modeled as proportional to the ball's

2. velocity. This means that the ball's acceleration, which is a derivative of its velocity, depends on
the velocity (and the velocity depends on time). Finding the velocity as a function of time
involves solving a differential equation and verifying its validity.
Types[edit]
Differential equations can be divided into several types. Apart from describing the properties of
the equation itself, these classes of differential equations can help inform the choice of approach
to a solution. Commonly used distinctions include whether the equation is: Ordinary/Partial,
Linear/Non-linear, and Homogeneous/Inhomogeneous. This list is far from exhaustive; there are
many other properties and subclasses of differential equations which can be very useful in
specific contexts.
Ordinary differential equations[edit]
Main article: Ordinary differential equation
An ordinary differential equation (ODE) is an equation containing a function of one independent
variable and its derivatives. The term "ordinary" is used in contrast with the termpartial
differential equation which may be with respect to more than one independent variable.
Linear differential equations, which have solutions that can be added and multiplied by
coefficients, are well-defined and understood, and exact closed-form solutions are obtained. By
contrast, ODEs that lack additive solutions are nonlinear, and solving them is far more intricate,
as one can rarely represent them by elementary functions in closed form: Instead, exact and
analytic solutions of ODEs are in series or integral form. Graphical and numerical methods,
applied by hand or by computer, may approximate solutions of ODEs and perhaps yield useful
information, often sufficing in the absence of exact, analytic solutions.