Symbolic math in Matlab

Matlab has some capability to perform symbolic math.

Contents

Solve the quadratic equation

syms a b c x  % this declares a b c and x to be symbolic variables

f = a*x^2 + b*x + c

solution = solve(f,x)

 
f =
 
a*x^2 + b*x + c
 
 
solution =
 
 -(b + (b^2 - 4*a*c)^(1/2))/(2*a)
 -(b - (b^2 - 4*a*c)^(1/2))/(2*a)
 

the solution you should recognize in the form of although Matlab does not print it this nicely!

pretty(solution) % this does a slightly better ascii art rendition

 
  +-                       -+ 
  |          2         1/2  | 
  |    b + (b  - 4 a c)     | 
  |  - -------------------  | 
  |            2 a          | 
  |                         | 
  |          2         1/2  | 
  |    b - (b  - 4 a c)     | 
  |  - -------------------  | 
  |            2 a          | 
  +-                       -+

differentiation

you might find this helpful!

diff(f) % first derivative
diff(f,2) % second derivative

 
ans =
 
b + 2*a*x
 
 
ans =
 
2*a
 

diff(f,a) % derivative of f with respect to a

 
ans =
 
x^2
 

integration

int(f)

int(f, 0, 1) % definite integral from 0 to 1

 
ans =
 
(a*x^3)/3 + (b*x^2)/2 + c*x
 
 
ans =
 
a/3 + b/2 + c
 

Analytically solve a simple ODE

ode = 'Dy = y';  % note the syntax: Dy = dydx
init = 'y(0)=1';
independent_variable = 'x';
y = dsolve(ode, init, independent_variable)

 
y =
 
exp(x)
 

evaluate the solution at a few places

say x=4 using the subs command.

subs(y,4)

ans =

   54.5982

we can also evaluate it on a range of values, say from 0 to 1.

x = 0:0.1:1;
Y = subs(y,x);
plot(x,Y)
xlabel('x values')
ylabel('y values')
title('Solution to the differential equation y''(x) = y(x)')

Warning! this note is in the dsolve documentation

Note By default, the solver does not guarantee general correctness and completeness of the results. If you do not set the option IgnoreAnalyticConstraints to none, always verify results returned by the dsolve command.

Let's see what this means

y1 = dsolve('Dy=1+y^2','y(0)=1')

y2 = dsolve('Dy=1+y^2','y(0)=1',...
'IgnoreAnalyticConstraints','none')

 
y1 =
 
tan(pi/4 + t)
 
 
y2 =
 
piecewise([C13 in Z_, tan(pi/4 + t + pi*C13)])
 

those solutions look different, but it isn't clear what do to with the second one.

see also: http://www.cs.utah.edu/~germain/PPS/Topics/Matlab/symbolic_math.html and http://www.mathworks.com/help/toolbox/symbolic/brvfu8o-1.html

% categories: Symbolic
% tags: math

Published with MATLAB® 7.10