Finding a Second Linearly Independent Solution Using Reduction of Order

[Mark Brewer]

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?

 

[SteamKing]

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.

 

[vela]

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.

 

[HallsofIvy]

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 y_2= u(x)x^{1/2}cos(x)
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?