Second Order ODE Homogenous

  • #1

Homework Statement


(Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution.

The Attempt at a Solution



I divided x2 to both sides to get the equation into y" + py' + qy = 0

y" + (1/x)y' + ((1 - (1/4x2)) = 0

Using Abel's method

c e-∫(p)dx

= c e-ln x
y2 = -c lnx[/B]

Am I doing this right?
 

Answers and Replies

  • #2
SteamKing
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Homework Statement


(Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution.

The Attempt at a Solution



I divided x2 to both sides to get the equation into y" + py' + qy = 0

y" + (1/x)y' + ((1 - (1/4x2)) = 0

Using Abel's method

c e-∫(p)dx

= c e-ln x
y2 = -c lnx[/B]

Am I doing this right?
Take y2 and plug it back into the original differential equation and see if it is satisfied. :wink:
 
  • #3
vela
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Homework Statement


(Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution.

The Attempt at a Solution



I divided x2 to both sides to get the equation into y" + py' + qy = 0

y" + (1/x)y' + ((1 - (1/4x2)) = 0

Using Abel's method

c e-∫(p)dx

= c e-ln x
y2 = -c lnx[/B]

Am I doing this right?
No, the problem said to use the method of reduction of order.

But first, did you type the differential equation and solution correctly? I ask because the supposed solution doesn't satisfy the given differential equation.
 
  • #4
HallsofIvy
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Homework Helper
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Homework Statement


(Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0.
You mean x2y" + xy' + (x2 - 1/4)y = 0, don't you? As Vela pointed out, the given y1 does not satisfy the equation you gave. Perhaps it satisfies this one. I haven't checked.

Use the method of reduction of order to find another linearly independent solution.
So you start by looking for a solution of the form [itex]y_2= u(x)x^{1/2}cos(x)[/itex]
Find the first and second derivatives of that and put them into the equation. If your given function really is a solution to the differential equation, then all terms involving only "u" (as opposed to u' or u'') will cancel leaving a first order equation for v= u'.

3. The Attempt at a Solution

I divided x2 to both sides to get the equation into y" + py' + qy = 0

y" + (1/x)y' + ((1 - (1/4x2)) = 0

Using Abel's method

c e-∫(p)dx

= c e-ln x
y2 = -c lnx
Am I doing this right?

Homework Statement


(Reduction of order) The function y1 = x-1/2cosx is one solution to the differential equation x2y" + xy' + (x2 - 1/4) = 0. Use the method of reduction of order to find another linearly independent solution.

The Attempt at a Solution



I divided x2 to both sides to get the equation into y" + py' + qy = 0

y" + (1/x)y' + ((1 - (1/4x2)) = 0

Using Abel's method

c e-∫(p)dx

= c e-ln x
y2 = -c lnx[/B]

Am I doing this right?
 

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