Having Trouble with this DEQ

  • Thread starter dsr39
  • Start date
In summary, troubleshooting a DEQ involves checking connections and consulting resources for solutions. Common issues include incorrect connections, power supply problems, software errors, and hardware malfunctions. To prevent future issues, proper storage and maintenance, regular updates, and avoiding extreme conditions are recommended. While some issues can be fixed independently, complex problems may require professional assistance. The frequency of calibration for a DEQ depends on the specific model and usage, with a general recommendation of at least once a year or when performance changes. Refer to the user manual for specific calibration guidelines.
  • #1
dsr39
14
0
I am trying to solve

[tex]x^2y'' + xy' = 0[/tex]

I know that two solutions that work, by inspection are

[tex]y_1 = c_1[/tex]

and

[tex]y_2 = c_2ln(x)[/tex]

where [tex]c_1[/tex] and [tex]c_2[/tex] are just arbitrary constants.

However I was hoping I would be able to find a more systematic way to solve it. I can get the constant solution by the method of Frobenius, but when I tried to get the natural log solution using the Wronskian, I wasn't able to get it.

I ended up with the second solution being the integral of a Gaussian or something of that sort. I know the Wronskian method is always supposed to work for DEQ's of this type. Where am I going wrong? Can someone show me how to use the Wronskian and the constant solution to get the natural log solution?

(Also why do my LaTeX expressions look funny)

Thanks for the help
 
Physics news on Phys.org
  • #2
Since y itself does not appear in that equation, there is a pretty standard method. Let u= y'. Then the equation becomes x2u'+ xu= 0 which is separable:
x2u'= -xu or (1/u)du= -(1/x)dx. Integrating, ln(u)= -ln(x)+ C or dy/dx= u= c/x (where c= eC) dy= (c/x)dx gives y= cln(x)+ D.

This also happens to be an "Euler-type" or "equi-potential" equation- the power of x in each term is equal to the order of the derivative. The change of variable, u= ln(x), so that x= eu, makes that an equation with constant coefficients: dy/dx= (dy/du)(du/dx)= (1/x)dy/du and d2y/dx2= d/dx(dy/dx)= d/dx((1/x)dy/du)= (1/x)d/du((1/x)dy/du)= (1/x)((1/x)d2y/du2)-(1/x2dy/dx.
The equation becomes d2y/dx2- dy/du+ dy/du= d2/du2= 0. Integrating once y'(u)= C. Integrating again y(u)= Cu+ D. Since u= ln(x), y(x)= Cln(x)+ D.
 
  • #3
dsr39 said:
(Also why do my LaTeX expressions look funny)

Hi dsr39! :smile:

Do you mean in the line "where [tex]c_1[/tex] and [tex]c_2[/tex] are just arbitrary constants."?

Because when it's in the middle of a text line, you need to use "itex" (for "inline latex") instead of "tex" …

where [itex]c_1[/itex] and [itex]c_2[/itex] are just arbitrary constants. :wink:
 
  • #4
hmm... What if the equation is (x^2)*y"[x]+x*y'[x]-y[x] = 0?

How different would the solution be?
 
  • #5
x^2*y''+x*y'=0
x*y''+y'=0
(x*y')'=0
x*y'=C1
y'=C1/x
y=C1*log(x)+C2
 
  • #6
Not sure if u are replying to my question or the original poster's question.
What if there is an additional y[x] term in the equation? i.e.

(x^2)*y"[x]+x*y'[x]-y[x] = 0
 
  • #7
We could try guessing solutions.

Try y = x. This works.

Indeed, y = cx works, for all constants c.

Does y = cx + d work? not really, since you have that 'd' left over.

So far, we can be sure that the general solution will contain a "cx" term.

Similarly, one can see that a/x works too; try it out.

So I'm guessing the general solution is

y(x) = ax + b/x

Notice that it has two degrees of freedom, as required for a 2nd order ODE.
 
  • #8
(x^2)*y"+x*y'-y = 0
[(x^2)*y'-x*y]'=0
let y=u/x
[(x^2)*y'-x*y]'=0->x*u''-u'=0
u'=2x*C2
u=C1+(x^2)*C2
y=u/x=C1/x+x*C2
 
Last edited:
  • #9
Nice.

Does it come out to be what I got?
 
  • #10
^Now that I fixed my error they match.
 

1. How do I troubleshoot a DEQ?

To troubleshoot a DEQ, first check if all necessary equipment is properly connected and powered on. Next, review the user manual or online resources for common issues and their solutions. If the problem persists, contact the manufacturer or a professional for further assistance.

2. What are some common issues with a DEQ?

Some common issues with a DEQ include incorrect connections, power supply issues, software errors, and hardware malfunctions. Additionally, improper use or settings can also cause problems.

3. How can I prevent future issues with my DEQ?

To prevent future issues with your DEQ, make sure to properly store and maintain the equipment. Regularly check for updates and install them as needed, and avoid using the DEQ in extreme environmental conditions.

4. Can I fix a DEQ on my own?

It depends on the issue and your level of expertise. Some simple issues, such as incorrect connections, can be fixed on your own. However, for more complex issues, it is best to seek professional help to avoid causing further damage.

5. How often should I calibrate my DEQ?

The frequency of calibration depends on the specific DEQ and its usage. Generally, it is recommended to calibrate the DEQ at least once a year or whenever there is a change in the equipment's performance. Refer to the user manual for specific guidelines on calibration for your DEQ.

Similar threads

  • Differential Equations
Replies
2
Views
884
Replies
3
Views
720
  • Calculus and Beyond Homework Help
Replies
2
Views
133
  • Calculus and Beyond Homework Help
Replies
5
Views
155
  • Differential Equations
Replies
1
Views
1K
  • Differential Equations
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
7
Views
412
  • Calculus and Beyond Homework Help
Replies
7
Views
606
  • Calculus and Beyond Homework Help
Replies
10
Views
388
  • Differential Equations
Replies
2
Views
2K
Back
Top