Separation of variables for second order DE

[daudaudaudau]

Hi. I believe I understand separation of variables for a first order DE. But can anyone tell me how to use it on a second order DE? In particular I have been looking at this example
http://en.wikipedia.org/wiki/Integrating_factor#General_use"
where it is claimed that one can use separation of variables to solve
$$\frac{d^2 y}{dt^2}=Ay^{2/3}$$

 

[arildno]

Hi. I believe I understand separation of variables for a first order DE. But can anyone tell me how to use it on a second order DE? In particular I have been looking at this example
http://en.wikipedia.org/wiki/Integrating_factor#General_use"
where it is claimed that one can use separation of variables to solve
$$\frac{d^2 y}{dt^2}=Ay^{2/3}$$

Multiply each side with dy/dt,and assuming that y(t) is defined&differentiable at t=0 (say, an initial value problem), we get, :
$$\frac{1}{2}(y'(t)^{2}-y'(0)^{2})=\frac{3A}{5}(y(t)^{\frac{5}{3}}-y(0)^{\frac{5}{3}})$$
Thus, you can get the separable diff.eq:
$$y'(t)=\pm\sqrt{Cy^{\frac{5}{3}}+D}$$
where the constants C and D can be determined from the above.
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[ross_tang]

First of all, for the 2nd order differential equation, the Wikipedia method is correct, but not easy to understand or use. Normally we would write:

$$\begin{align*} \frac{d^2y}{dt^2}&=\frac{dy'}{dt} \\ &=\frac{dy'}{dy} \frac{dy}{dt}\\ &= y' \frac{dy'}{dy} \end{align*}$$

And the general method in solving this type of equations is taught in this how-to: http://www.voofie.com/content/115/solving-2nd-order-ordinary-differential-equation-of-special-form-yt-fy/"

And for separation of variables, I think you have misunderstood a little bit. For the 1st order DE, Wikipedia used the method of integrating factor. Separation of variables refers to moving two different variables in different side, and do the integration. For instance, for the 2nd order DE:

Moving from:
$$y' \frac{d y'}{d y}=A y^{\frac{2}{3}}$$

To:
$$\int y' d y'=\int A y^{\frac{2}{3}}dy + C$$

We have used the separation of variables. For 1st order DE:

These 3 steps:
$$f(x) \frac{d y}{d x}=g(y)$$
$$\frac{1}{g(y)} \frac{d y}{d x}=\frac{1}{f(x)}$$
$$\int \frac{d y}{g(y)}=\int \frac{d x}{f(x)}+C$$
is separation of variables as well.