1. Symbolic Solution of differential equations

This is the purely symbolic solution of an differential equation, obtained by anti-differentiation:

>	ODE := x*diff(y(x),x) = y(x)*ln(x*y(x)) - y(x);

>	sol := dsolve(ODE,y(x));

This is a first-order ODE, we derived only once with respect to x.  The result is a family parametrized by the complex constant C1.  To verify whether we have a solution, we substitute the solution into the ODE.  Conveniently that the solution is an equation, because we can give this as first argument to subs:

>	subs(sol,ODE);

>	simplify(%);

This is a nice illustration of the normalization problem in symbolic computation.

2. Series Expansions to solve

If a symbolic solution does not exist as an explicit formula, we may still compute symbolically an approximate answer, using series.

>	diffeq := L*diff(theta(t),t$2) = -g*sin(theta(t));

This ODE expresses the motion of a pendulum, made from a mass hanging from a string of length L, subject to the gravitation (constant g).  In order to have a formal solution, we simplify sin(theta) = theta, for small values of theta.

We turn this problem into an Initial Value Problem (IVP) by the conditions

>	inits := theta(0) = 0, D(theta)(0) = v[0];

The two initial conditions express that at t = 0, we have no deviation in the angle, but we have an initial velocity v[0].  Notice the introduction of the "D" operator, here we consider theta as a function, whereas in the differential equation, we use the "diff" operator.

>	IVP := {diffeq,inits};

>	sol := dsolve(IVP,theta(t),series);

This is a formal solution, we see the parameters L, g, v[0] as symbols.  But it is also an approximate solution, valid only for small values of t, t << 1.

3. Numerical Solutions

To illustrate the numerical option, we take another differential equation:

>	diffeq := diff(y(t),t$2) - (1-y(t)^2)*diff(y(t),t) + y(t) = 0;

>	inits := y(0) = 0, D(y)(0) = -0.1;

>	IVP := {diffeq,inits};

>	sol := dsolve(IVP,y(t),numeric);

Maple returns a procedure, which we may evaluate, for any value of t, say t = 0:

>

sol(0);

On return we get a list of equations.  First the value of t, then the position y(t), and then the velocity.  Here we recognize our initial conditions.

>

sol(1);

To plot the solution, we often create an extra function:

>	ysol := t -> rhs(sol(t)[2]);

>

ysol(1);

>	plot(ysol,0..20);

>

4. Graphical Solutions

We can plot a vector field of trajectories, for systems of first order equations.

See the handout to convert this problem into a system of two first order equations, introducting an extra variable yy(t) for diff(y(t),t).