### Differential Equations

``` # Maple solves differential equations using dsolve:

deq := diff( y(x), x\$2 ) + y(x) = 0;   # y''(x)+y(x)=0
dsolve( deq, y(x) );

#  Initial conditions give a unique solution that can be graphed.

de := diff( y(x), x ) = y(x) - sin(x);
ic := y(0)=01;
sol:= dsolve( {de,ic}, y(x));

# To graph the solution, extract the expression after the equal sign
# of the answer. Maple function rhs does this, and then the expression
# can be graphed.

u:=rhs(sol);
plot( u, x = 0..10);

# Let's make a slope field, using  the DEplot command from
# the DEtools package:

with(DEtools):
de := diff( y(x), x ) = y(x) - sin(x);
DEplot( de, y(x), x = -3..3, y = -3..3,
arrows = line, dirgrid = [30,30]);

# Plot a few trajectories and the direction field.

ics:=[ [y(0)=0], [y(0)=1], [y(0)=3] ];
# Set of initial conditions. A double list.
# Identical Result: ics:=[[0,0],[0,1],[0,3]]:
opts:=color='black',linecolor='red',arrows='line',dirgrid=[30,30]:
DEplot( de, y(x), x = -3..3, y = -3..3,opts,ics);
```