Solving xy' + 2y = sqrt(1 + x^2)

[cscott]

Homework Statement

Solve xy' + 2y = sqrt(1 + x^2)

Homework Equations

I got an integrating factor of $x^2$

So $$y = 1/x^2 \int{x\sqrt{1 + x^2}}$$

The Attempt at a Solution

and I get $$y =\frac{(1 + x^2)^{3/2}}{3x^2}$$ for C = 0

But this wrong when I substitute it in ... what'd I do wrong?

 

[cscott]

How would I solve this in Maple? When I do and try it's answer it seems wrong as well... it tells me [itex]2y = \sqrt(1+x^2)[/tex]

 

[arildno]

It is wrong because you have multiplied the sqruare root expression with x^2, rather than with just x:

$$xy'+2y=\sqrt{1+x^{2}}\to{x}^{2}y'+2xy=x\sqrt{1+x^{2}}\to(x^{2}y)'=x\sqrt{1+x^{2}}$$
And so on..

 

[cscott]

sorry, I that was a typo... with $x\sqrt{1 + x^2}$ you get the y as above. (at least I do... on my calculator too) but when I substitute for y into the LHS it doesn't equal $\sqrt{1+x^2}$

 

[D H]

I just used the Maple packaged with Matlab.

>> syms x y C1

>> ysoln = dsolve('x*Dy+2*y=sqrt(1+x^2)','x')
ysoln = (1/3*(1+x^2)^(3/2)+C1)/x^2

>> x*diff(ysoln,x)+2*ysoln-sqrt(1+x^2)
ans = A big awful mess

>> pretty(simple(ans))
0​

In other words, the answer you got in the original post is correct.

Moral: Sometimes Maple (or Mathematica) create a big long mess. Simplification cleans it up, sometimes (as it did in this case).

Usually when I use Maple I am left with a big long mess that Maple can't simplify. I use Maple to check my hand work for stupid errors.

>>mysoln = some simple equation;
>>maplesoln = some big long mess;
>>zerocheck = simple(mysoln-maplesoln)​

Maple doesn't make the stupid mistakes I am quite prone to make. On the other hand, Maple is quite stupid about simplifying things.

 

[joob]

I agree with DH that 'dsolve' is a good tool to use to solve these diffs on maple.
There are also a lot of great tools in maple to simplify answers. If all your getting on maple are big nasty answers, your not using it right.
read up on 'simplify' ,'factor', and 'collect', those are the 3 most common tools i use

 

[cscott]

Thanks... for the tips too!

 

[D H]

There are also a lot of great tools in maple to simplify answers. If all your getting on maple are big nasty answers, your not using it right.
read up on 'simplify' ,'factor', and 'collect', those are the 3 most common tools i use

Don't be insulting. Those tools only go so far. The bottom line is that simplification is not particularly well-defined and is definitely not easy (it is an NP-hard problem). I have run across many problems that neither Maple nor Mathematica can simplify as well as a human can. I have seen many, many more examples in autogenerated code (no human would write a multi-page equation).

 

[joob]

Sorry, I didnt mean to sound insulting.