MATLAB: Differential System Of Equations

Question

I am trying to find answers to specific input (t=20)

But my code keeps out putting strange equations without all the variable

I'd appreciate it if anyone can show me why this output is like that

For example the first line below is part of the output I dont understand what the #X is

xSol(t) =
 
(exp(root(#X^3 + (5*#X)/4 - 2^(1/2)/8, #X, 3)*t)*(2^(1/2)
clc, clear
syms x(t) y(t) z(t)
ode1 = diff(x,t) == z - (1/2)*y;
ode2 = diff(y,t) == (1/2)*x  - (1/(sqrt(2)))*z;
ode3 = diff(z,t) == (1/(sqrt(2)))*y - (1/2)*x;
odes = [ode1; ode2; ode3];
cnd1 = x(0) == 1;
cnd2 = y(0) == 0;
cnd3 = z(0) == 0;
conds = [cnd1; cnd2; cnd3];
[xSol(t), ySol(t), zSol(t)] = dsolve(odes, conds)
fplot(xSol)
hold on
fplot(ySol)
hold on
fplot(zSol)
grid on
legend('xSol','ySol', 'zSol')

Answer 1

No mystery at all.

Add these three assignments after your dsolve call:

xSolvpa = vpa(xSol)
ySolvpa = vpa(ySol)
zSolvpa = vpa(zSol)

and the functions resolve to:

xSolvpa(t) =
0.39703503403676880167413394762852*exp(0.13926074738154957998312484530008*t) + exp(t*(- 0.069630373690774789991562422650042 - 1.1245199717305828384502799984853i))*(0.30148248298161559916293302618574 - 0.0059166222071075799552646099750859i) + exp(t*(- 0.069630373690774789991562422650042 + 1.1245199717305828384502799984853i))*(0.30148248298161559916293302618574 + 0.0059166222071075799552646099750859i)
 
ySolvpa(t) =
0.32349030593970774096648972843982*exp(0.13926074738154957998312484530008*t) - exp(t*(- 0.069630373690774789991562422650042 - 1.1245199717305828384502799984853i))*(0.16174515296985387048324486421991 - 0.19227126160807200590439356187362i) - exp(t*(- 0.069630373690774789991562422650042 + 1.1245199717305828384502799984853i))*(0.16174515296985387048324486421991 + 0.19227126160807200590439356187362i)
 
zSolvpa(t) =
0.21703654854647326974599442155993*exp(0.13926074738154957998312484530008*t) - exp(t*(- 0.069630373690774789991562422650042 - 1.1245199717305828384502799984853i))*(0.10851827427323663487299721077996 + 0.24247546582044825480541269698659i) - exp(t*(- 0.069630373690774789991562422650042 + 1.1245199717305828384502799984853i))*(0.10851827427323663487299721077996 - 0.24247546582044825480541269698659i)

If you want to use them numerically, use the matlabFunction (link) function. That will convert them to anonymous functions.

EDIT —

To evaluate them and plot them:

x20 = xSolvpa(20)                               % Evaluate At t=20


y20 = ySolvpa(20)                               % Evaluate At t=20
z20 = zSolvpa(20)                               % Evaluate At t=20
fplot(xSolvpa, [0 30])
hold on
fplot(ySolvpa, [0 30])
hold on
fplot(zSolvpa, [0 30])
grid on
legend('xSol','ySol', 'zSol', 'Location','N')

They evaluate to complex results, although you can safely ignoire the imaginary parts, since they are vanishingly small:

x20 =
6.3031762131597174425074106544313 + 1.1479437019748901445007192746311e-41i
 
y20 =
5.2664281481456048004144813508306 - 2.2958874039497802890014385492622e-41i
 
z20 =
3.6217244010970352709039208520563