# Solving an ODE ( Analytically - General Solution )
# example y’=tan(y)/(x-1)
y = function ('f')(x) # defines the solution as a function
de = diff (y,x) == cos(x)-cot(x)*y # defines the differential equation
h = desolve (de , y) # solves analytically ( when possible )
show (h) # shows implicit solution
show (solve (h,y)) # shows solution if its a function
# This is the general solution as we did not give initial conditions

# section 2.1 number 28
# Solving an IVP ( Analytically - Specific Solution )
# example y’=cos(x)-cot(x)*y y(pi/2) =1
y = function ('f')(x) # defines the solution as a function
de = diff (y,x) == cos(x)-cot(x)*y # defines the differential equation
h = desolve (de , y, ics =[pi/2,1]) # solves analytically ( when possible )
show (h) # shows right hand side of solution
show (solve (h,y)) # shows solution if its a function
# This is the particular solution as we gave initial conditions

#3 question
#plotting graph for above solution
# Plotting a sequence of solution curves with a direction field for an ODE
# example y'+2*y=-x^2*e^(-2*x) which has solutions y=e^(-2*x)*(x^4/4+C)
x,y= var('x,y')
I=sum( plot (-(cos(x)^2-2*c)/(2*sin(x)),(x,.5,2.0)) for c in range (-5,10)) #varies the constant c in the general solution y=e^( -2*x)*(x ^4/4+ c)
D= plot_slope_field(cos(x)-2*cot(x)*y ,(x,0.5,2.0),(y , -10 ,15),headaxislength=3, headlength =3) # plots the direction field (solve for y' and put in right hand side)
P= plot (-(cos(x)^2-2)/(2*sin(x)),(x,0.5,2.0),color='orange', thickness=3) #emphasizes a specific solution in orange
show (I+D+P)