# Second Order Homogeneous Differential Equation from Optimal Controls

Please excuse my moment of ignorance while I reboot my education in math. I am taking an optimal controls course and it has been quite some time since I've worked with calculus.

On to my question...

I have a 2nd order homogeneous differential equation that represents an optimal solution as follows:

$$tx''(t)-3x'(t) = 0$$

For those who are not familiar with controls notation, this equation can be written in classical $x$-$y$ format as such:

$$xy''-3y' = 0$$

What is throwing me off here is the non-constant coefficient before $y''$. I do not remember how to go about solving this and wikipedia is only confusing me further.

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This is actually a first order differential equation in disguise. Let $u = y'$ to get:

$$x u' - 3 u = 0$$

This is a separable equation:

$$\frac{u'}{u} = \frac{3}{x}$$

Integrate:

$$\log u = 3 \log x + a_1 = \log x^3 + a_1$$

Therefore:

$$u = \exp(\log x^3 + a_1) = b_1 x^3$$

Where $b_1 = \pm \exp(a_1)$.

Given that $u = y'$, we have:

$$y' = b_1 x^3$$

Integrate to get the solution:

$$y = c_1 x^4 + c_2$$

Where $c_1 = b_1/4$.

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Hi Ayman. Thanks for the edit and response! Can you explain the part between step 2 and 3? – Kashif Oct 2 '12 at 19:55
@Kashif We just integrate both sides. You know that the integral of $u'/u$ is $\log u$, right? – Ayman Hourieh Oct 2 '12 at 20:11
No, I don't remember that. I remember that the derivative of $log(u)$ is $1/u$ so then the integral of $u'/u$ would be something else entirely, I think – Kashif Oct 2 '12 at 20:37
@Kashif The derivative of $\log(u)$ is $1/u$ if $u$ is an independent variable. If $u$ is a function of $x$, then the derivative of $\log(u)$ is $u'/u$ by the chain rule. – Ayman Hourieh Oct 3 '12 at 0:40
Ah, okay. I see it now. Thank you! – Kashif Oct 3 '12 at 0:44