# Thread: Finding the Second Order Linear Homogeneous Equation from the given Fundamental Pair

1. ## Finding the Second Order Linear Homogeneous Equation from the given Fundamental Pair

The problem instructions read like so:

Show that the given functions are linearly independent on the interval I and ﬁnd
a second-order linear homogeneous equation having the pair as a fundamental set of
solutions.

$\displaystyle y_1=x$
$\displaystyle y_2=x^2$
$\displaystyle I=(\infty,-\infty)$

I'm able to prove that they are linearly independent easily by simply using the Wronskian Matrix to get

$\displaystyle W(x)=3x^3\neq0\in I$

However, I don't know how to find the equation based off of what I'm given.
I know the answer to the question (it's in the book) so I will post it for reference of what I'm looking for.

$\displaystyle x^2 y\prime\prime - 2x y\prime +2y = 0$

please show me the method to find this equation based off of the fundamental set given!

2. Originally Posted by Jenkins
The problem instructions read like so:

Show that the given functions are linearly independent on the interval I and ﬁnd
a second-order linear homogeneous equation having the pair as a fundamental set of
solutions.

$\displaystyle y_1=x$
$\displaystyle y_2=x^2$
$\displaystyle I=(\infty,-\infty)$

I'm able to prove that they are linearly independent easily by simply using the Wronskian Matrix to get

$\displaystyle W(x)=3x^3\neq0\in I$

However, I don't know how to find the equation based off of what I'm given.
I know the answer to the question (it's in the book) so I will post it for reference of what I'm looking for.

$\displaystyle x^2 y\prime\prime - 2x y\prime +2y = 0$

please show me the method to find this equation based off of the fundamental set given!
You should be able to set up three equations with three unknowns:

$\displaystyle a_2(x)y''_1+a_1(x)y'_1+a_o(x)y_1=0$

$\displaystyle a_2(x)y''_2+a_1(x)y'_2+a_o(x)y_2=0$

$\displaystyle a_2(x)(y_1+y_2)''+a_1(x)(y_1+y_2)'+a_o(x)(y_1+y_2) =0$

Try that...

You should be able to set up three equations with three unknowns:

$\displaystyle a_2(x)y''_1+a_1(x)y'_1+a_o(x)y_1=0$

$\displaystyle a_2(x)y''_2+a_1(x)y'_2+a_o(x)y_2=0$

$\displaystyle a_2(x)(y_1+y_2)''+a_1(x)(y_1+y_2)'+a_o(x)(y_1+y_2) =0$

Try that...
I tried it and got that
$\displaystyle a_2=0$
$\displaystyle a_1=0$
$\displaystyle a_o=0$

which makes sense considering equations 1 and 2 both equal 0 and equation 3 is a linear combination of the two.

4. Yeah, I was just about to edit that. It doesn't work.

5. Ok, I think I know what to do now, lol. I was on the right track... Just define $\displaystyle P(x)=\frac{a_1(x)}{a_2(x)}$ and $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$, basically divide the first two equations (from my failed system) by the coefficient of y'' to get:

$\displaystyle y_1''+P(x)y'_1+Q(x)y_1=0$

$\displaystyle y_2''+P(x)y'_2+Q(x)y_2=0$

From the first:

$\displaystyle P(x)=-Q(x)x$

Now substitute into the second:

$\displaystyle 2-Q(x)2x^2+Q(x)x^2=0$

$\displaystyle Q(x)=\frac{2}{x^2}$

That should help...

Remember we defined $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$.

Ok, I think I know what to do now, lol. I was on the right track... Just define $\displaystyle P(x)=\frac{a_1(x)}{a_2(x)}$ and $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$, basically divide the first two equations (from my failed system) by the coefficient of y'' to get:

$\displaystyle y_1''+P(x)y'_1+Q(x)y_1=0$

$\displaystyle y_2''+P(x)y'_2+Q(x)y_2=0$

From the first:

$\displaystyle P(x)=-Q(x)x$

Now substitute into the second:

$\displaystyle 2-Q(x)2x^2+Q(x)x^2=0$

$\displaystyle Q(x)=\frac{2}{x^2}$

That should help...

Remember we defined $\displaystyle Q(x)=\frac{a_o(x)}{a_2(x)}$.
you are magical