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I'm recently figuring out how to make this equation solved but Mathematica does not solve this??!!

DSolve[{ y'[t]== ((3 a)/2 (y[t] - b/(2 a))^2 + k ((3 a)/2 - 1) t^-2 - ((a f)/2 + (3 b^2)/(8 a)))/(t y[t] ), y[1] == y0}, y, t]

a,b, k and f are the positive parameters.

Thank guys

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  • $\begingroup$ You can format inline code and code blocks by selecting the code and clicking the {} button above the edit window. The edit window help button ? is useful for learning how to format your questions and answers. You may also find the meta Q&A, How to copy code from Mathematica so it looks good on this site, helpful $\endgroup$
    – Michael E2
    Nov 19 '21 at 19:07
  • $\begingroup$ I changed it in Inputform to see better. $\endgroup$
    – Mathecis
    Nov 19 '21 at 19:44
  • $\begingroup$ It's still not formatted right — at least before, it could be copied and pasted directly into Mathematica. But now it's a different equation than at first. $\endgroup$
    – Michael E2
    Nov 19 '21 at 19:48
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    $\begingroup$ It can be solved for b = 0, but probably not for other values. Any reason to think it can be solved symbolically? $\endgroup$
    – Michael E2
    Nov 19 '21 at 19:49
  • $\begingroup$ for b=0 it gets easy but I have to consider b. $\endgroup$
    – Mathecis
    Nov 19 '21 at 19:54
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There does not seem to have a closed form analytical solution. The next best thing (other than a numerical solution) is to find a series solution by expanding around $t=1$

ClearAll[y, t, a, b, k, f]
ode = t*y[t]*y'[t] + (3*a)/2*(y[t] - b/(2*a))^2 + 
    k*((3*a)/2 - 1)*t^(-2) - ((a*f)/2 + (3*b^2)/(8*a)) == 0;
AsymptoticDSolveValue[{ode, y[1] == y0}, y[t], {t, 1, 4}]

Mathematica graphics

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  • $\begingroup$ That's a very good idea, also I was thinking about how we can plot this on the surface (2D) to see if there are some singularities?!! $\endgroup$
    – Mathecis
    Nov 20 '21 at 13:37
  • $\begingroup$ I would like to plot this like that: $\endgroup$
    – Mathecis
    Nov 20 '21 at 16:53
  • $\begingroup$ @Mathecis to plot the above solution, just need to define numerical values for all the unknown parameters, y0,a,b,k,f and after that you can use the plot command on the result. $\endgroup$
    – Nasser
    Nov 20 '21 at 17:38
  • $\begingroup$ Is no any method to see an analytical way?? $\endgroup$
    – Mathecis
    Nov 24 '21 at 15:59
  • $\begingroup$ @Mathecis I do not think there is no closed form solution. series method is the next best thing. But you can ask at the math forum if you want. $\endgroup$
    – Nasser
    Nov 24 '21 at 16:10

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