#section 2.2 number 10
y= function('y')(x) # defines the solution as a function
de = (y-1)^2*diff(y,x) == 2*x + 3  #defines the differential equation
h = desolve (de , y) # solves analytically ( when possible )
show(h)

# Plotting a sequence of integral curves with a direction field for ODE
# example y'=(2*x+1)/(5*y^4+1) which has the implicit solutions y^5+y=x^2+x+C
x,y= var('x,y')
I=sum(implicit_plot ((1/3)*y^3-y^2+y == x^2+3*x+0.5*c, (x ,-2 ,2) , (y , -2 ,2)) for c in range ( -10 ,30)) # varies the constant c in the general solution y^5+y=x^2+ x+c
D= plot_slope_field ((2* x+3) /(y-1)^2 , (x ,-2 ,2) , (y , -2 ,2) , headaxislength =3,headlength =3) # plots the direction field
P= implicit_plot ((1/3)*y^3-y^2+y == x^2+3*x, (x ,-2 ,2) , (y , -2 ,2) , color ='orange') # emphasizes a specific solution in orange
show (I+D+P) # plots all three on one graph