Slove the series ,the equation (x2-1)d^2y/dx^2+3xdy/dx+ xy=0 in power of (x-2)
Answers
Answer:
Introduction
A differential equation is an equation relating an independent variable, e.g. t, a dependent variable, y,
and one or more derivatives of y with respect to t:
dx
dt
= 3x y2 dy
dt
= e
t d
2y
dx
2
+ 3x
2
y
2 dy
dx
= 0.
In this section we will look at some specific types of differential equation and how to solve them.
2 Classifying equations
We can classify our differential equation by four properties:
• Is it an ordinary differential equation?
• Is it linear?
• Does it have constant coefficients?
• What is the order?
Ordinary
An Ordinary Differential Equation or ODE has only one independent variable (for example, x, or t).
The alternative (with more than one) is called a partial differential equation and will not be covered in
this course.
Linearity
A differential equation is linear if every term in the equation contains none or exactly one of either the
dependent variable or its derivatives. There are no products of the dependent variable with itself or its
derivatives. Each term has at most one power of the equivalent of x or ˙x or ¨x or . . . ; or f(x) and its
derivatives.
Examples:
f(x)
df
dx
= −ω
2x is not linear df
dx
= f
3
(x) is not linear d
2f
dx
2
= −x
2
f(x) + e
x
is linear.
Constant coefficients
A differential equation has constant coefficients if the dependent variable and all the derivatives are only
multiplied by constants.
Examples: which have constant coefficients?
3
df
dx
= −ω
2x: yes
d
2f
dx
2
= −x
2
f(x) + e
x
: no
d
2f
dx
2
+ 3
df
dx
+ 2f(x) = sin xex
: no.