Variation of parameters

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Homework Statement



x^2y''-2xy'+2y=x^3cosx

Find a general solution by using variation of parameters

2. The attempt at a solution

I already solved this one, but I have 4 questions:

1. I found the solutions x and x^2 to the homogeneous equation by inspection. Is this the only way to do it?

2. How about the solutions -x and -x^2? I first tried the solutions x^2 and -x^2, and that gave me 0 as the particular solution...Why can't I choose these two?

3. When calculating the Wronski, how do I know which solution is y1 and y2? I first chose y1 as x^2, and that gave me a different solution...

4. Why are there no constants of integration?
 

Answers and Replies

  • #2
HallsofIvy
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Homework Statement



x^2y''-2xy'+2y=x^3cosx

Find a general solution by using variation of parameters

2. The attempt at a solution

I already solved this one, but I have 4 questions:

1. I found the solutions x and x^2 to the homogeneous equation by inspection. Is this the only way to do it?
The homogeneous equation here is [itex]x^2 y"- 2xy'+ 2=0[/itex]. That's an "equipotential" equation, also called an "Euler type" equation. You can get its general solution by "trying" y= xr as a solution. Also the substitution x= ln(t) changes the equation into a "constant coefficients" equation that is easy to solve.

2. How about the solutions -x and -x^2? I first tried the solutions x^2 and -x^2, and that gave me 0 as the particular solution...Why can't I choose these two?
The general solution to the homogeneous equation is Cx+ Dx2 where C and D can be any contstants. You can choose them to be -1, certainly. I don't know what you mean by "gave me 0 as the particular solution". y identically equal to 0 certainly doesn't satisfy the equation.

3. When calculating the Wronski, how do I know which solution is y1 and y2? I first chose y1 as x^2, and that gave me a different solution...
The Wronskian is a determinant. If you swap two columns, you just multiply the Wronskian by -1. That should do no more than multiply your previous solution by -1.

4. Why are there no constants of integration?
Because youdidn't put them in! Your general solution should be Cx+ Dx2+ a particular solution to the entire equation.
 
  • #3
HallsofIvy
Science Advisor
Homework Helper
41,847
967

Homework Statement



x^2y''-2xy'+2y=x^3cosx

Find a general solution by using variation of parameters

2. The attempt at a solution

I already solved this one, but I have 4 questions:

1. I found the solutions x and x^2 to the homogeneous equation by inspection. Is this the only way to do it?
The homogeneous equation here is [itex]x^2 y"- 2xy'+ 2=0[/itex]. That's an "equipotential" equation, also called an "Euler type" equation. You can get its general solution by "trying" y= xr as a solution. Also the substitution x= ln(t) changes the equation into a "constant coefficients" equation that is easy to solve.

2. How about the solutions -x and -x^2? I first tried the solutions x^2 and -x^2, and that gave me 0 as the particular solution...Why can't I choose these two?
The general solution to the homogeneous equation is Cx+ Dx2 where C and D can be any contstants. You can choose them to be -1, certainly. I don't know what you mean by "gave me 0 as the particular solution". y identically equal to 0 certainly doesn't satisfy the equation.

3. When calculating the Wronski, how do I know which solution is y1 and y2? I first chose y1 as x^2, and that gave me a different solution...
The Wronskian is a determinant. If you swap two columns, you just multiply the Wronskian by -1. That should do no more than multiply your previous solution by -1.

4. Why are there no constants of integration?
Because youdidn't put them in! Your general solution should be Cx+ Dx2+ a particular solution to the entire equation.
 
  • #4
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Thanks!
 
Last edited:
  • #5
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The Wronski keeps confusing me, however.

W(x^2, x) = -x^2

W(x, x^2) = x^2

The particular solution depends on which one of these I choose.
 
  • #6
382
0
The Wronski keeps confusing me, however.

W(x^2, x) = -x^2

W(x, x^2) = x^2

The particular solution depends on which one of these I choose.

Or not?
 

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