Why Can We Multiply Both Sides of a Differential Equation by u(x)?

[flyingpig]

Homework Statement

My book did this

y' + p(x)y = q(x)

u(x)y' + u(x)p(x)y = u(x)q(x)

Then they did some algebra and product rule manipulation and turned it into a seperatable diff eqtn

Now here is my problem, why is it that they can multiply both sides by u(x)? doesn't that change the whole thing completely? I know that they can always "cancel it" but this still changes the whole function

also this works if and only if u'(x) = p(x)u(x)

but my book also assumed that

du(x) = p(x)u(x)dx

du(x)/u(x) = p(x)dx

ln|u(x)| = integral of p(x) dx

Now here is the problem, how do we know u(x) is linear? What happens if u(x) is a trig function??

It's like saying

y = x

xy = x²

y = x²/x

but x can't be 0 now.

 

[SammyS]

Hello !

I'll try to cover the main points. This will not directly answer all of your questions.

The point of having u'(x) = p(x)u(x), is that then the equation

u(x)y' + u(x)p(x)y = u(x)q(x)​

becomes

u(x)y' + u'(x)y = u(x)q(x) .​

This is handy because the product rule gives

$$\frac{d}{dx}(u(x)·y)=u(x)\frac{dy}{dx}+\left(\frac{d(u(x))}{dx}\right)y$$

$$=u(x)y' + u'(x)y$$​

This allows differential equation

u(x)y' + u(x)p(x)y = u(x)q(x)​

to be written as

$$\frac{d}{dx}(u(x)·y)=u(x)q(x)\ .$$​

This equation can be solved (for u(x)y) by integrating both sides w.r.t x . You then solve this for y by dividing both sides of the resulting equation by u(x), so of course u(x) must not be the zero function.

u(x) is not just an arbitrary function you pick. It has to be such that u'(x)=p(x)u(x).

 

[Ray Vickson]

If L(x) is the left-hand side and R(x) is the right-hand side, the equation L(x) = R(x) remains true or false when both sides are multiplied by any u(x). That is, if we happen to have a value of x for which L = R is true, the relation u*L = u*R will also be true at that x. After all, if A = 5 then 3*A = 3*5. It is TRUE that 3*A is not A and 3*5 is not 5, but the _relationship_ remains true. Similarly, if L = R happens to be false at some x, it remains false when both sides are multiplied by the same _non-zero_ value u. After all, if 4 is different from 7 then 5*4 is different from 5*7.

The book does not claim that u*(y' + p*y) is the same as y' + p*y and it does not claim that u*q is the same as q. It just says that the two sides remain equal.

RGV