# A No-linear differential equation

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

• 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 Nov 19 '21 at 19:07
• I changed it in Inputform to see better. Nov 19 '21 at 19:44
• 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. Nov 19 '21 at 19:48
• It can be solved for b = 0, but probably not for other values. Any reason to think it can be solved symbolically? Nov 19 '21 at 19:49
• for b=0 it gets easy but I have to consider b. Nov 19 '21 at 19:54

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}]


• 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?!! Nov 20 '21 at 13:37
• I would like to plot this like that: Nov 20 '21 at 16:53
• @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. Nov 20 '21 at 17:38
• Is no any method to see an analytical way?? Nov 24 '21 at 15:59
• @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. Nov 24 '21 at 16:10