Qfyilyi

DSolve gives a strange solution for the Riccati differential equation $ y' = (y^2) - 2 x^2 y + (x^4) + 2 x + 4 $

Opres = DSolve[y'[x] == y[x]^2-2x^2*y[x]+x^4+2x+4, y[x], x]

$left{left{y(x)to frac{1}{c_1 e^{4 i x}-frac{i}{4}}+x^2-2 iright}right}$

When I try plot this solution

Opresgraf = 

 Plot[Evaluate[y[x] /. Opres /. C[1] -> Range[-3, 3]], {x, -4.7, 4.7}, 

  PlotRange -> 4.7]

I get a blank graph.

My question is: how can I get a solution with DSolve (not with NDSolve, because in my student research project I need DSolve) and plot that solution, the most important is to plot that general solution with DSolve.

perhaps

Plot[Evaluate[ReIm@y[x] /. (Opres /. C[1] -> Range[-3, 3])], {x, -4.7, 4.7}, 

  PlotRange -> 4.7]

Try this

Opres = DSolve[y'[x] == y[x]^2-2x^2 *y[x]+x^4+2x+4, y[x], x][[1]];

Plot[{Re[y[x]/.Opres/.C[1]->Range[3.3]],Im[y[x]/.Opres/.C[1]->Range[3.3]]}, {x,-4.7,4.7}] 

With a single graph you can only plot those solution that are imaginary or real.

There are 2 real ones:

sol = First[DSolve[y'[x] == y[x]^2 - 2 x^2*y[x] + x^4 + 2 x + 4, y[x], x]];

zeroIm = FullSimplify[ComplexExpand[Im[y[x] /. sol]]] == 0 // Solve[#, C[1]] &

$left{left{C[1]to -frac{1}{4}right},left{C[1]to frac{1}{4}right}right}$

Plot[Evaluate[y[x] /. sol /. zeroIm], {x, -4.7, 4.7}]