MATLAB: Substitute only first derivative in symbolic toolbox

derivativefirstsecondsubstitutesymbolicSymbolic Math Toolbox

I am considering a second order differential equation (mass – spring system), and I want to substitute the first derivative with another symbolic function.

syms x(t) v(t) m k
expr = m * diff(x, t, 2) + k * x(t) == 0;
expr2 = subs(expr, diff(x, t, 1), v(t));

Anyway, result of expr2 still containts the second derivative of x(t) and not the first derivative of v(t). Is there a way I can solve it and make it explicit?

>> expr2
expr2(t) =
m*diff(x(t), t, t) + k*x(t) == 0

Best Answer

I remember doing vibration analysis, used to be pretty tricky:

 expr2 = subs(expr, diff(x, 2), diff(v))

Related Solutions

MATLAB: How to symbolically work with implicit functions (i.e., “x(t)”) within the Symbolic Math Toolbox

The following example demonstrates how you can represent "x" as functions of time "t" and perform differentation on it:

Example 1:

syms  t
x = sym('x(t)'); % x(t)
E1 = diff(x) * sin(x) 
E2 = diff(E1,t)
pretty(E1)
pretty(E2)

The solution for E1:

                               /d      \
                               |-- x(t)| sin(x(t))
                               \dt     /

The solution for E2:

                   / 2      \
                   |d       |             /d      \2
                   |--- x(t)| sin(x(t)) + |-- x(t)|  cos(x(t))
                   |  2     |             \dt     /
                   \dt      /

The following example demonstrates how to substitute 'x(t)' with an explicit definition.

Example 2:

% Order of the derivative
n = 2
% Define symbolic variables.
syms t
x = sym('t^2');
f = sym('f(x)');
% SUBS is used to substitute definition of x
f_subs = subs(f);
disp('Diff f_subs wrt to t')
pretty(diff(5*f_subs + t^2,t,n))
disp(' ')
disp('Diff f_subs wrt to x(t)')
pretty(diff(5*f_subs + t^2,x,n))
disp(' ')

MATLAB: I can’t plot 2nd order differential equation

I can!

Try this:

syms x(t);
Dx = diff(x);
Dx2 = diff(x,2);
x(t) = dsolve(Dx2+(5*Dx)+(6*x)==100,x(0)==0,Dx(0)==0)
figure(1)
fplot(x, [0 10])
axis([xlim    0  17])
grid

Best Answer

Related Solutions

MATLAB: How to symbolically work with implicit functions (i.e., “x(t)”) within the Symbolic Math Toolbox

MATLAB: I can’t plot 2nd order differential equation

Related Question