> May I suggest you test ODE with mathHand.com? It can solve many ODE that other cannot. e.g. DrHuang.com/index/bug
As far as I can tell, your bug list are all variations of the same bug that has, at some point, been fixed. Here are five examples from your list in Mathematica 13.1...
In[160]:= DSolve[y'[x] - y[x]^2 - 2 x^2 - 1 == 0, y[x], x]
Out[160]= {{y[
x] -> -(((1 + I) 2^(1/4) C[
1] (((1 +
I) x ParabolicCylinderD[-(1/4)
I (-2 I + Sqrt[2]), (1 + I) 2^(1/4) x])/2^(3/4) -
ParabolicCylinderD[
1 - 1/4 I (-2 I + Sqrt[2]), (1 + I) 2^(1/4) x]) - (1 -
I) 2^(1/4) (-(((1 - I) x ParabolicCylinderD[
1/4 I (2 I + Sqrt[2]), (-1 + I) 2^(1/4) x])/2^(3/4)) -
ParabolicCylinderD[
1 + 1/4 I (2 I + Sqrt[2]), (-1 + I) 2^(1/4) x]))/(C[
1] ParabolicCylinderD[-(1/4) I (-2 I + Sqrt[2]), (1 + I) 2^(
1/4) x] +
ParabolicCylinderD[
1/4 I (2 I + Sqrt[2]), (-1 + I) 2^(1/4) x]))}}
In[161]:= DSolve[y'[x] - 2 y[x]^2 - 2 x^2 - 1 == 0, y[x], x]
Out[161]= {{y[
x] -> -((2 (-1)^(1/4) C[
1] ((-1)^(1/4)
x ParabolicCylinderD[-(1/2) - I/2, 2 (-1)^(1/4) x] -
ParabolicCylinderD[1/2 - I/2, 2 (-1)^(1/4) x]) +
2 (-1)^(3/
4) ((-1)^(3/4)
x ParabolicCylinderD[-(1/2) + I/2, 2 (-1)^(3/4) x] -
ParabolicCylinderD[1/2 + I/2, 2 (-1)^(3/4) x]))/(2 (C[
1] ParabolicCylinderD[-(1/2) - I/2, 2 (-1)^(1/4) x] +
ParabolicCylinderD[-(1/2) + I/2, 2 (-1)^(3/4) x])))}}
In[163]:= DSolve[y'[x] - 3 y[x]^2 - 2 x^2 - 1 == 0, y[x], x]
Out[163]= {{y[
x] -> -(((1 + I) 6^(1/4) C[
1] (((1 + I) 3^(1/4)
x ParabolicCylinderD[-(1/4)
I (-2 I + Sqrt[6]), (1 + I) 6^(1/4) x])/2^(3/4) -
ParabolicCylinderD[
1 - 1/4 I (-2 I + Sqrt[6]), (1 + I) 6^(1/4) x]) - (1 -
I) 6^(1/4) (-(((1 - I) 3^(1/4)
x ParabolicCylinderD[
1/4 I (2 I + Sqrt[6]), (-1 + I) 6^(1/4) x])/2^(3/4)) -
ParabolicCylinderD[
1 + 1/4 I (2 I + Sqrt[6]), (-1 + I) 6^(1/4) x]))/(3 (C[
1] ParabolicCylinderD[-(1/4)
I (-2 I + Sqrt[6]), (1 + I) 6^(1/4) x] +
ParabolicCylinderD[
1/4 I (2 I + Sqrt[6]), (-1 + I) 6^(1/4) x])))}}
In[165]:= DSolve[y''[x] - y'[x] y[x] - x == 0, y[x], x]
Out[165]= {{y[
x] -> -((2 ((-1)^(
3/4) (1/2 (-1)^(3/4)
x ParabolicCylinderD[-(1/2) I (-I + C[1]), (-1)^(3/4)
x] - ParabolicCylinderD[
1 - 1/2 I (-I + C[1]), (-1)^(3/4) x]) + (-1)^(1/4) C[
2] (1/2 (-1)^(1/4)
x ParabolicCylinderD[1/2 I (I + C[1]), (-1)^(1/4) x] -
ParabolicCylinderD[
1 + 1/2 I (I + C[1]), (-1)^(1/4)
x])))/(ParabolicCylinderD[-(1/2) I (-I + C[1]), (-1)^(
3/4) x] +
C[2] ParabolicCylinderD[1/2 I (I + C[1]), (-1)^(1/4) x]))}}
In[168]:=
DSolve[y'[x]^2 - x y'[x] - y[x] == 0, y[x], x] // FullSimplify
Out[168]= {{y[
x] -> (-4 E^(3 C[1])
x - (-2 x^2 + (-E^(6 C[1]) - 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) - 8 x^3)^3])^(1/3))^2)/(
8 (-E^(6 C[1]) - 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) - 8 x^3)^3])^(1/3))}, {y[x] ->
1/16 (8 x^2 + ((4 + 4 I Sqrt[3]) x (E^(3 C[1]) + x^3))/(-E^(
6 C[1]) - 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) - 8 x^3)^3])^(
1/3) + (1 - I Sqrt[3]) (-E^(6 C[1]) - 20 E^(3 C[1]) x^3 +
8 x^6 + Sqrt[E^(3 C[1]) (E^(3 C[1]) - 8 x^3)^3])^(1/3))}, {y[
x] -> 1/16 (8 x^2 + ((4 - 4 I Sqrt[3]) x (E^(3 C[1]) + x^3))/(-E^(
6 C[1]) - 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) - 8 x^3)^3])^(
1/3) + (1 + I Sqrt[3]) (-E^(6 C[1]) - 20 E^(3 C[1]) x^3 +
8 x^6 + Sqrt[E^(3 C[1]) (E^(3 C[1]) - 8 x^3)^3])^(1/3))}, {y[
x] -> (4 E^(3 C[1])
x - (-2 x^2 + (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(1/3))^2)/(
8 (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(1/3))}, {y[
x] -> ((-4 - 4 I Sqrt[3]) E^(3 C[1]) x + (4 + 4 I Sqrt[3]) x^4 +
8 x^2 (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(
1/3) + (1 - I Sqrt[3]) (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 +
8 x^6 + Sqrt[E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(
2/3))/(16 (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(1/3))}, {y[
x] -> (4 I (I + Sqrt[3]) E^(3 C[1]) x + (4 - 4 I Sqrt[3]) x^4 +
8 x^2 (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(
1/3) + (1 + I Sqrt[3]) (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 +
8 x^6 + Sqrt[E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(
2/3))/(16 (-E^(6 C[1]) + 20 E^(3 C[1]) x^3 + 8 x^6 + Sqrt[
E^(3 C[1]) (E^(3 C[1]) + 8 x^3)^3])^(1/3))}}