Solving Differential Equations: Techniques and Methods

[der.physika]

Okay, so I was sitting in my room wondering about the differential operator $$D$$. Like for example, solving the equation

$$y\prime\prime+5y\prime+4y=0$$

introduce $$D=\frac{d}{dx}$$

$$(D^2+5D+4)y=0$$

so you can solve it by doing $$D=-4$$, $$D=-1$$

$$e^-^4^x+e^-^x$$ is the solution

But how the hell do you solve? $$y\prime\prime+2xy=0$$

or (with other variables involved?)

$$xy\prime+y=e^x^y$$

or $$2yy\prime\prime=(y\prime)^2$$

what is the method? What are the techniques?

 

[HallsofIvy]

You don't. Linear equations with variable coefficients require completely different techniques and often "new" functions. For example, the general solution to "Bessel's equation of order 0": $x^2y''+ xy'+ x^2y= 0$ involve the Bessel functions of the first and second kind (which are defined as solutions to Bessel's equation).

The special case of "equi-potential" or "Euler type" equations, which are of the form $ax^2y"+ bxy'+ cy= 0$, the coefficient of each derivative involving simply x to the same power as the degree of the derivative, can be changed to "constant coefficients" equations by the change of variable t= ln(x).

But other than that, the most general methods for linear equations with variable coefficients involve infinite sums: Assume a solution of the form $y(x)= \sum_{n=0}^\infty a_nx^n$, put that and its derivatives into the equation and try to get a recursive equation for $a_n$.

 

[epenguin]

There are as mentioned series methods. There are numerical methods that give you numerical small step by small step solutions that computers can plot out for you as curves etc. if you give them input initial conditions and numerical constants of the d.e. when necessary. Very necessary for practical applications. But you often like the analytical solutions you are calling for, formulae, for qualitative insight, or further mathematical treatment - others may explain you motives.Then in the books you find there is only one way to solve a differential equation. That is to know the answer! You do know how to differentiate? There is a general method for that. So if you do know the solution you can check that it is a solution. Or at least if you know the outline of the solution you can do the differentiations and knock the constants or ill-ftting bits into shape till you know you have the right answer. What they teach you in d.e. courses is, by examples, to excercise your eye till you realize you do already know the solution to some at first sight unrecognisable things. Or else how to hammer them till they become in a form you recognise. It is intellectually despicable and some of it is a fraud but parts of it are useful and it has a horrible fascination.

E.g. your second one below

But how the hell do you solve? $$y\prime\prime+2xy=0$$

or (with other variables involved?)

$$xy\prime+y=e^x^y$$

or $$2yy\prime\prime=(y\prime)^2$$

what is the method? What are the techniques?

because I have learned to look for that sort of thing, I recognise is $$(xy)'=e^x^y$$

i.e. let $$(xy)'=w$$ and it is $$w'=e^w$$ which you probably can solve, or rather know the solution of, already.

Your last example took a bit more hammering till I recognised y = x² is a solution.
I could probably do the first but I felt I should control this vice.

I'd suggest many students, say of chemistry, biology, economics and even physics if they are not going to have to solve d.e.'s in their profession would save themselves trouble by not following proceedures by which the books and profs. pretend to solve them, but going straight to the solutions given and checking them. It's just as good if you only want to understand the physics. Then perhaps working back see what they're doing.

Well maybe not quite 100% of Profs. will agree with me,

and I will say there is a bit of doctrine about linear d.e.s with constant coefficients that is best to make part of you. Grasp of this is also useful in qualitative analysis of d.e.'s that you can't actually solve analytically - particularly useful in mathematical biology.

I have heard echos of and can vaguely intuit a more general approach than the standard ragbag, involving Lie Groups, but it is not at all so widely known, and it is reasonable to suppose that if it were very practical it would be widely known.