Definition of variable coefficients in linear differential equations

[tiredryan]

Does the variable coeffcient in linear differential equations have to be polynomials? All the examples I have found seem to be polynomials for example the Cauchy–Euler equation.

For example is sin(x)y'' + cos(x^2)y' + y = 0 still considered a linear differential equation with variable coefficients?

http://en.wikipedia.org/wiki/Linear_differential_equation#Equation_with_variable_coefficients

 

[Studiot]

The way to tell if it is linear or not is to look at the functions (including derivatives) of y.
The equation is linear in y if all functions of y are in the first power.

That is y'', y', y occur but (y'')² ; (y')² ; (y)² do not occur.

The general second order linear differential equation is

$$l\left( x \right)\frac{{{d^2}y}}{{d{x^2}}} + m\left( x \right)\frac{{dy}}{{dx}} + n\left( x \right)y = g\left( x \right)$$

Where l(x), m(x), n(x) and g(x) are functions of x only.

So your example is linear.

go well

 

[Some Pig]

Yes, it is linear.

 

[HallsofIvy]

Yes, if we have a linear differential equation with variable coefficients with dependent variable y and independent variable x, then the coefficients may be any functions of x.

Of course, even with polynomial coefficents, such an equation can become very difficult to solve.

 

[atseek]

Hi everybody,
I just wanted to ask how can I solve such problems when the coefficient is a sin or cos?
What method should I use? My problem is

y'''+y''+y'+cos(x)y=0

and I don't even know how to start solving it. I was thinking I could replace cos with its power series and then take its laplace transform but that's way too long. I don't know if it would work anyway. Please help.
P.S. I didn't include the non-homogeneous term.