Find the explicit solution of the linear DE dy/xdx=-6/x^3-6x^2 y+1 using the appropriate integrating factor and with the initial value of y(10)=-15. (b)Find the largest interval of definition I for x. (c)Which of the terms in the solution are transient(show limits)? Solution A differential equation is a mathematicalequation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, and other disciplines. Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space and/or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time varies. Newton\'s laws allow one to relate the position, velocity, acceleration and various forces acting on the body and state this relation 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 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 deceleration due to air resistance. Gravity is constant but air resistance may be modelled as proportional to the ball\'s velocity. This means the ball\'s acceleration, which is the derivative of its velocity, depends on the velocity. Finding the velocity as a function of time involves solving a differential equation. Differential equations are mathematically studied from several different perspectives, mostly concerned with their solutions—the set of functions that satisfy the equation. Only the simplest differential equations admit solutions given by explicit formulas; however, some properties of solutions of a given differential equation may be determined without finding their exact form. If a self- contained formula for the solution is not available, the solution may be numerically approximated using computers. The theory of dynamical systems puts emphasis on qualitative analysis of systems described by differential equations, while many numerical methods have been developed to determine solutions with a given degree of accuracy. The term homogeneous differential equation has several distinct meanings. One meaning is that a first-order ordinary differential equation is homogeneous (of degree 0) if it has the form \\frac{dy}{dx} = F(x,y) where F(x,y) is a homogeneous function of degree zero; that is to say, such that F(tx,ty) = F(x,y). In a related, but distinct, usage, the term linear .

1. Find the explicit solution of the linear DE dy/xdx=-6/x^3-6x^2 y+1 using the appropriate
integrating factor and with the initial value of y(10)=-15. (b)Find the largest interval of definition
I for x. (c)Which of the terms in the solution are transient(show limits)?
Solution
A differential equation is a mathematicalequation for an unknown function of one
or several variables that relates the values of the function itself and its derivatives of various
orders. Differential equations play a prominent role in engineering, physics, economics, and
other disciplines. Differential equations arise in many areas of science and technology,
specifically whenever a deterministic relation involving some continuously varying quantities
(modeled by functions) and their rates of change in space and/or time (expressed as derivatives)
is known or postulated. This is illustrated in classical mechanics, where the motion of a body is
described by its position and velocity as the time varies. Newton's laws allow one to relate the
position, velocity, acceleration and various forces acting on the body and state this relation 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 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 deceleration due to air resistance.
Gravity is constant but air resistance may be modelled as proportional to the ball's velocity. This
means the ball's acceleration, which is the derivative of its velocity, depends on the velocity.
Finding the velocity as a function of time involves solving a differential equation. Differential
equations are mathematically studied from several different perspectives, mostly concerned with
their solutions—the set of functions that satisfy the equation. Only the simplest differential
equations admit solutions given by explicit formulas; however, some properties of solutions of a
given differential equation may be determined without finding their exact form. If a self-
contained formula for the solution is not available, the solution may be numerically
approximated using computers. The theory of dynamical systems puts emphasis on qualitative
analysis of systems described by differential equations, while many numerical methods have
been developed to determine solutions with a given degree of accuracy. The term homogeneous
differential equation has several distinct meanings. One meaning is that a first-order ordinary
differential equation is homogeneous (of degree 0) if it has the form frac{dy}{dx} = F(x,y)
where F(x,y) is a homogeneous function of degree zero; that is to say, such that F(tx,ty) = F(x,y).
In a related, but distinct, usage, the term linear homogeneous differential equation is used to
describe differential equations of the form Ly = 0 , where the differential operatorL is a linear

2. operator, and y is the unknown function. The remainder of this article is about homogeneous
differential equations in the first sense defined above. Solving homogeneous differential
equations By the definition above, it can be seen that F(tx,ty) = F(x,y) for all t, so t can be
arbitrarily chosen to simplify the form of the equation. One can solve this equation by making a
simple change of variables y = ux, and then using the product rule on the left hand side as
follows, frac{d(ux)}{dx} = xfrac{du}{dx} + ufrac{dx}{dx} = xfrac{du}{dx} + u. and
then using the identity F(tx,ty) = F(x,y) to simplify the right hand side by choosing to set t to be
1/x, transforming the original problem into the separable differential equation xfrac{du}{dx} +
u = F(1,u) which can then be integrated by the usual methods. Ordinary differential equation In
mathematics, an ordinary differential equation (or ODE) is a relation that contains functions of
only one independent variable, and one or more of their derivatives with respect to that variable.
A simple example is Newton's second law of motion, which leads to the differential equation m
frac{d^2 x(t)}{dt^2} = F(x(t)),, for the motion of a particle of constant mass m. In general, the
force F depends upon the position x(t) of the particle at time t, and thus the unknown function
x(t) appears on both sides of the differential equation, as is indicated in the notation F(x(t)).
Ordinary differential equations are distinguished from partial differential equations, which
involve partial derivatives of functions of several variables. Ordinary differential equations arise
in many different contexts including geometry, mechanics, astronomy and population modelling.
Many famous mathematicians have studied differential equations and contributed to the field,
including Newton, Leibniz, the Bernoulli family, Riccati, Clairaut, d'Alembert and Euler. Much
study has been devoted to the solution of ordinary differential equations. In the case where the
equation is linear, it can be solved by analytical methods. Unfortunately, most of the interesting
differential equations are non-linear and, with a few exceptions, cannot be solved exactly.
Approximate solutions are arrived at using computer approximations (see numerical ordinary
differential equations). Existence and uniqueness of solutions There are several theorems that
establish existence and uniqueness of solutions to initial value problems involving ODEs both
locally and globally. See Picard–Lindelöf theorem for a brief discussion of this issue.