Integral and ODE

59
0
1. Problem with Integral

I got stuck on the following Integal and can't find the mistake:

[tex]Integral(x/(x-1))[/tex]

let U' = x and V = (x-1)^(-1)
and U= 0.5x^2 V' = 1/(x-1)^2

so: [tex]x^2/(2(x-1))-1/2Integral(x^2/(x-1)^2[/tex]

algebraic division: [tex]x^2 : (x^2 -2x + 1) = 1- 1/(x-1)^2 + 1/(x-1)^2[/tex]

substituting back into equation gives:
[tex] 1/2 [x^2/(x-1) - x - ln (x-1)^2 x 1/(x-1)] = (x+1)/(2(x-1)) + ln(x-1)[/tex]

but the result is supposed to be:
[tex]x + ln(x-1)[/tex]

2. Substitution in ODE

[tex]xydy/dx + (x^2 + y^2 +x) = 0 [/tex]

I have tried to substitute z = xy and also z = x^2 + y^2, but that didn't work. Any other ideas?

Thank you!
 
Last edited:
59
0
PS: How can I write integrals with the [tex] function?
 
T

Tx

Guest
For the first, Plus and Minus one from the top. This will give:
I (x - 1 + 1)/ (x - 1) dx
I 1dx + I 1/(x-1) dx
x + ln(x-1) + C
#

The second, I got:
y^2 = x(1-x)/(1+x) + C
However, I am new to ODE's so it's probably wrong.
 
59
0
*lol* thanks... that was easy.... I seem to have a tendency to make things more complicated than they actually are...:redface:

But thanks again!
 
BTW, to do integrals in TeX:

\int f(x) dx
[tex]\int f(x) dx[/tex]

\int_0^5 f(x) dx
[tex]\int_0^5 f(x) dx[/tex]
 

saltydog

Science Advisor
Homework Helper
1,579
2
Hey mmy, the second one IS complicated. Just for the record in case you haven't figured it out, make it exact by finding an integrating factor. You know partial this, partial that, some arithmetic, get x. Solving, I get:

[tex]y(x)=\pm\frac{\sqrt{\frac{k}{x^2}-3x^2-4x}}{\sqrt{6}}[/tex]

with the sign dictated by the initial conditions.
 
I got

[tex]
y(x) = \frac{1}{x}\sqrt{C - \frac{1}{2}x^4 - \frac{2}{3}x^3}
[/tex]
 
twoflower said:
I got
[tex]
y(x) = \frac{1}{x}\sqrt{C - \frac{1}{2}x^4 - \frac{2}{3}x^3}
[/tex]
Which is the same as saltydog has, I see :)
 

The Physics Forums Way

We Value Quality
• Topics based on mainstream science
• Proper English grammar and spelling
We Value Civility
• Positive and compassionate attitudes
• Patience while debating
We Value Productivity
• Disciplined to remain on-topic
• Recognition of own weaknesses
• Solo and co-op problem solving
Top