> with(DEtools);

[DEnormal, DEplot, DEplot3d, Dchangevar, PDEchangecoords, PDEplot,

    autonomous, convertAlg, convertsys, dfieldplot, indicialeq,

    phaseportrait, reduceOrder, regularsp, translate, untranslate,

    varparam]

> dsolve({sin(x)*diff(y(x),x)+y*cos(x)=x*sin(x),y(Pi/2)=2},y(x));

                            -sin(x) + x cos(x) - 1
                   y(x) = - ----------------------
                                    sin(x)

> dsolve({diff(y(x),x)+y*cos(x)/sin(x)=x,y(Pi/2)=2},y(x));

                            -sin(x) + x cos(x) - 1
                   y(x) = - ----------------------
                                    sin(x)

> yofx:=rhs(");

                             -sin(x) + x cos(x) - 1
                   yofx := - ----------------------
                                     sin(x)

> plot(yofx,x=1/2..5/2);

# Why not plot it from 0 to 10? Because there is a singularity in the
# original differential equation corresponding to x=0 or x=Pi, and there
# is also a singularity in the solution, division by zero, at those
# values of x. So a prudent plot command will stand back from those
# points. 
# 
# Now let's see what happens when we call upon DEplot to give us some
# more insight. 
> DEplot(diff(y(x),x)+y*cos(x)/sin(x)=x,y(x),x=1/4..3,y=0..4);

# What?! This is nothing like the down-then-up behavior of the formal
# solution. 
> DEplot(diff(y(x),x)+y*cos(x)/sin(x)=x,y(x),x=1/4..3,[[y(Pi/2)=2]],y=0.
> .4);

# The solution does not run parallel to the arrows of the direction
# field here, either.
# 
# This simply cannot be right. Your OWN insight must tell you that this
# is wrong. 
> ?DEplot
# It seems that although the form in which we've been stating the task
# does not provoke outright rebellion from the machine, it provokes the
# subtler and more dangerous false compliance. That is, you get a plot
# but it's wrong. 
> DEplot(diff(y(x),x)=-y*cos(x)/sin(x)+x,y(x),x=1/4..3,[[y(Pi/2)=2]],y=0
> ..4);

# Ah, that's much better. 
# 
# "STANDARD FORM" for DEplot, requires that the differential equation be
# put in the form 
# 
# diff(y(x),x)=formula in terms of x and y. 
# 
# This is a different standard form, than the form we use to work with
# first order linear differential equations, for instance.