I am trying to solve the following ode in Maple. Here is the code below 

dsolve(diff(b(x), x, x, x)/2 + diff(b(x), x, x)/x + 2*diff(b(x), x)/(x^2 + 1)^2 - 4*x*b(x)/(x^2 + 1)^3 = 0)

and the output I get is the following:

b(x) = DESol({-8*x^2*_Y(x) + (4*x^3 + 4*x)*diff(_Y(x), x) + (2*x^6 + 6*x^4 + 6*x^2 + 2)*diff(_Y(x), x, x) + (x^7 + 3*x^5 + 3*x^3 + x)*diff(_Y(x), x, x, x)}, {_Y(x)})

I don't understand what the function _Y(x) means. Could someone please explain? 

Thanks in advance!  

I am working with the de Sitter metric which takes the form 

$$g=\tau^{-2}\left(-\left(\Lambda / 3-\tau^{2}\right)^{-1} d \tau^{2}+\left(\Lambda / 3-\tau^{2}\right) d t^{2}+g_{\mathbb{S}^{2}}\right)$$

or 

g := evalDG((-`dτ` &t `dτ`/(Lambda/3 - tau^2) + (Lambda/3 - tau^2)*(dt &t dt) + (dtheta &t dtheta) + sin(theta)^2*(dphi &t dphi))/tau^2)

in the co-ordinates $\{\tau, t, \theta, \phi\}.$

or 

DGsetup([tau, t, theta, phi], M1, verbose)

Define

F := evalDG(-Q*(`dτ` &t dt) + Q*(dt &t `dτ`))

Then I want to define the following function which takes a two tensor (metric) and returns a two tensor as well. 

Here \dot(g) is the input tensor, F is the two tensor defined above and the g is the de Sitter metric. 

I want to evaluate this expression of \dot{g} = d tau^2/tau^2 and other metrics. How can I do this in Maple?

Any help will be much appreciated.

I have the following ode: 

ode1 := diff(x(t), t, t) = (5*9.80665)*sin((1/6)*Pi)-(10*(10-sqrt(x(t)^2+25)))*x(t)/sqrt(x(t)^2+25)-(diff(x(t), t))

I tried the following code: 

DEplot(ode, x(t), t = -2 .. 2, [`$`([x(0) = (1/4)*k], k = -20 .. 20)], x = -8 .. 8, color = blue, stepsize = 0.5e-1, linecolour = red, arrows = MEDIUM)

But I get the following error: 

Error, (in DEtools/DEplot/CheckInitial) too few initial conditions: [x(0) = -5]

Any help in plotting this differential equation will be much appreciated. 

I could keep on using the commands 

with(GraphTheory):

with(RandomGraphs):

DrawGraph(RandomTree(7))

but then this is not helpful because I do not get non-isomorphic graph each time and there are repetitions. So any other suggestions would be very helpful. 

I have to first plot the some point like (-1,4), (0,3), (1,4) etc. and then I wish to know how I can connect them with a line.