First-order equations causing some trouble (for newbie)

[Pzi]

Hi.

So I'm new to differential equations (it has been a month since I started attending this course).

\[3{x^2} - y = y'\sqrt {{x^2} + 1} \]

http://img641.imageshack.us/img641/9833/eqn7464.png

A linear equation, quite obviously... However I'm pretty much not able to proceed due to complicated integration.
And I expected something reasonably elegant...

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\[5y + {(y')^2} = x(x + y')\]

http://img36.imageshack.us/img36/4629/eqn8105.png

I tried to solve for y' , which seems to be

\[y' = \frac{{x \pm \sqrt {5{x^2} - 20y} }}{2}\]

http://img99.imageshack.us/img99/3154/eqn6514.png

But once more I am not able to proceed...

 

[HallsofIvy]

Yes, that's just not a good way to proceed. Not every differential equation can be immediately reduced to an integral.

You are correct that this equation is linear which means it is relatively easy to find an "integrating factor".

We can write the equation as $y'\sqrt{x^2+ 1}+ y= 3x^2$. That is equivalent to $y'+ (x^2+1)^{-1/2} 2x^2(x^2+1)^{-1/2}$

Now we look for a function, $\mu(t)$, such that
$$\mu(x)(y'+ (x^2+1)^{1/2}y)= \mu(x) y'+ \mu(x)(x^2+1)^{-1/2}y= \frac{d(\mu y)}{dx}$$
($\mu$ is the "integrating factor".)

Since, by the product rule, $d(\mu(x)y)/dx= \mu y'+ \mu'y$ we must have $\mu'= (x^2+1)^{-1/2}/mu$ which is a separable equation for $\mu$:
$$\frac{d\mu}{\mu}= (x^2+ 1)^{-1/2}dx$$

 

[Pzi]

Yes, that's just not a good way to proceed. Not every differential equation can be immediately reduced to an integral.

You are correct that this equation is linear which means it is relatively easy to find an "integrating factor".

We can write the equation as $y'\sqrt{x^2+ 1}+ y= 3x^2$. That is equivalent to $y'+ (x^2+1)^{-1/2} 2x^2(x^2+1)^{-1/2}$

Now we look for a function, $\mu(t)$, such that
$$\mu(x)(y'+ (x^2+1)^{1/2}y)= \mu(x) y'+ \mu(x)(x^2+1)^{-1/2}y= \frac{d(\mu y)}{dx}$$
($\mu$ is the "integrating factor".)

Since, by the product rule, $d(\mu(x)y)/dx= \mu y'+ \mu'y$ we must have $\mu'= (x^2+1)^{-1/2}/mu$ which is a separable equation for $\mu$:
$$\frac{d\mu}{\mu}= (x^2+ 1)^{-1/2}dx$$

Thanks for response!

$$\begin{array}{l}<br /> \frac{{d\mu }}{\mu } = \frac{{dx}}{{\sqrt {{x^2} + 1} }}\\<br /> \ln \left| \mu \right| = \ln \left| {x + \sqrt {{x^2} + 1} } \right|\\<br /> \mu = x + \sqrt {{x^2} + 1} <br /> \end{array}$$

Which means

$$\begin{array}{l}<br /> y' + \frac{y}{{\sqrt {{x^2} + 1} }} = \frac{{3{x^2}}}{{\sqrt {{x^2} + 1} }}\\<br /> y' \cdot (x + \sqrt {{x^2} + 1} ) + \frac{{y(x + \sqrt {{x^2} + 1} )}}{{\sqrt {{x^2} + 1} }} = \frac{{3{x^2}(x + \sqrt {{x^2} + 1} )}}{{\sqrt {{x^2} + 1} }}\\<br /> \left( {y \cdot (x + \sqrt {{x^2} + 1} )} \right)' = \frac{{3{x^3}}}{{\sqrt {{x^2} + 1} }} + 3{x^2}\\<br /> y \cdot (x + \sqrt {{x^2} + 1} ) = {x^3} + \int {\frac{{3{x^3}dx}}{{\sqrt {{x^2} + 1} }}} <br /> \end{array}$$

And I'd be good to go from here. Can you confirm?