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I am attempting to make a plot of the solution an ODE. I want to plot the solution using PolarPlot, x is the polar angle in this case. I cannot get Mathematica to make a plot, and I'm not sure where I'm going wrong. I tried going about it this way:

sol = DSolve[r'[x] == Sqrt[-r[x]^2 - 2*A*r[x]^5 + 3*A^[2/3]*r[x]^4], r[0] == 0, r[x], x]
PolarPlot[sol, {x, 0, 8*Pi}] 

but nothing is produced on the graph. Is this wrong?

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You cannot ignore the error messages Mathematica puts out and then expect it to work. A^[2/3] is not a valid expression. The expression must be put in curly brackets if you provide a boundary condition. DSolve provides the result as a Rule you cannot plot a rule straight up. Look at the documentation for DSolve and study the examples. – Matariki Nov 18 '13 at 4:14
Other than syntax errors, you can't hope to "plot" something if your "A" has no numerical value. Mathematica needs to know the numerical value of "A" to plot the solution. – Nasser Nov 18 '13 at 4:43

If I make an assumption about the value of A, I can get a solution although it's not very pretty. First I solved your ODE with no boundary condition.

sol = With[{A = 2}, 
  DSolve[r'[x] == Sqrt[-r[x]^2 - 2*A*r[x]^5 + 3*A^(2/3)*r[x]^4], r[x], x]]


Next I extract an expression that can plotted from the result.

r[x_] = (sol /. x_C -> 0)[[1, 1, 2]];

Finally I make a plot.

Plot[r[x], {x, -1.3, 0.2}]


This isn't a polar plot. The domain and range of r[x] is so restricted that I don't think a polar plot displays it well. I further think that it is likely there is something wrong with your formulation of your ODE, but knowing nothing of the background to your problem, this is mere speculation.

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The differential equation has constraints on real solutions and the initial condition of r[0]==0 is problematic.

Insights can be obtained by the following:

  1. Checking when derivative has real values:

    w[a_] := N@Reduce[-y^2 + 3 a^(2/3) y^4 - 2 a y^5 >= 0, y]
  2. Plotting the function:

    Manipulate[Plot[Sqrt[-y^2 + 3 a^(2/3) y^4 - 2 a y^5], {y, -2, 2}], {a, 0.05,

    enter image description here

  3. Exploring the shape and domain of real solutions using StreamPlot:

    Manipulate[StreamPlot[{1, Sqrt[-y^2 + 3 a^(2/3) y^4 - 2 a y^5]}, {x, -4, 
    4}, {y, -2, 0}], {a, 0.05, 20}]

enter image description here

These issues have contributed to difficulties plotting solutions (as well as syntax errors and failure to specify value for A referred to in comments).

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