Page 56 - MATLAB an introduction with applications

MATLAB Basics ——— 41

1.19.2 Solution to Differential Equations
Symbolic math functions can be used to solve a single equation, a system of equations and differential
equations. For example:
solve( f ) : Solves a symbolic equation f for its symbolic variable. If f is a symbolic expression, this function
solves the equation f = 0 for its symbolic variable.
solve(f1, …, fn): Solves the system of equations represented by f 1, …, f n.
The symbolic function for solving ordinary differential equation is dsolve as shown below:
dsolve(‘equation’, ‘condition’): Symbolically solves the ordinary differential equation specified by ‘equation’.
The optional argument ‘condition’ specifies a boundary or initial condition.
The symbolic equation uses the letter D to denote differentiation with respect to the independent variable.
D followed by a digit denotes repeated differentiation. Thus, Dy represents dy/dx, and D2y represents
2
2
d y/dx . For example, given the ordinary second order differential equation;
2
dx + 5 dx + x 7
3 =
dt 2 dt
with the initial conditions x(0) = 0 and (0)x = 1.
The MATLAB statement that determines the symbolic solution for the above differential equation is the
following:
x = dsolve(‘D2x = –5*Dx – 3*x + 7’, ‘x(0) = 0’, ‘Dx(0) =1’)
The symbolic functions are summarized in Table 1.37.
Table 1.37 Solution of equations
compose Functional composition
dsolve Solution of differential equations
finverse Functional inverse
solve Solution of algebraic equations

1.19.3 Calculus
There are four forms by which the symbolic derivative of a symbolic expression is obtained in MATLAB.
They are:
diff( f ) : Returns the derivative of the expression f with respect to the default independent variable.
diff( f, ‘t’) : Returns the derivative of the expression f with respect to the variable t.
diff( f, n) : Returns the nth derivative of the expression f with respect to the default independent variable.
diff( f,‘t’, n) : Returns the nth derivative of the expression f with respect to the variable t.

The various forms that are used in MATLAB to find the integral of a symbolic expression f are given and
summarized in Table 1.38.

int( f ) : Returns the integral of the expression f with respect to the default independent variable.
int( f, ‘t’) : Returns the integral of the expression f with respect to the variable t.
int( f, a, b) : Returns the integral of the expression f with respect to the default independent variable
evaluated over the interval [a, b], where a and b are numeric expressions.
int(f,‘t’, a, b): Returns the integral of the expression f with respect to the variable t evaluated over the
interval [a, b], where a and b are numeric expressions.

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