sfpn=StreamPlot[{1,x*(y)^(1/2)},{x,-4,4},{y,-1,7},Axes-> Automatic,VectorStyle->Arrowheads[0.01]]

gensoln=DSolve[y'[x]==x*(y[x])^(1/2),y[x],x];

f[x_]=Simplify[gensoln[[1,1,2]]]

f[ Sqrt[-2*C[1]]]

f'[Sqrt[-2*C[1]]]

Simplify[f'[x]-x*(f[x])^(1/2)]

LHS=Integrate[y^(-1/2),y]

RHS=Integrate[x,x]+C

Reduce[RHS≥0,x]

LHS-RHS

Solve[LHS-RHS==0,y]

partsoln0=DSolve[{y'[x]==x*((y[x]))^(1/2), y[0]==0},y[x],x];

g[x_]=partsoln0[[1,1,2]]

g'[0]

grg=Plot[g[x],{x,-2,2}]

partsoln1=DSolve[{y'[x]==x*((y[x]))^(1/2), y[1]==0},y[x],x];

h1[x_]=Simplify[partsoln1[[1,1,2]]]

h[x_]:=If[x≥1||x≤-1,h1[x],0]

h'[1]

grh=Plot[h[x],{x,-2,2}]

Show[grg,grh]

Simplify[g'[x]-x*(g[x])^(1/2)]

Simplify[h1'[x]-x*(h1[x])^(1/2)]