Differential System Of Equations

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

[Star Strider]

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

Answer 2

[madhan ravi]

ic = [1;0;0]; % initial conditions
tspan = [0 20];
[t,x]=ode45(@myod,tspan,ic);  % function call
when_t_is_20 =  x(t==20,:) % solution 1 , 2 & 3
figure
plot(t,x,'-o')
h=legend('x(t)','y(t)','z(t)');
h.FontSize=20;

function dxdydz = myod(t,x) % function definition (save it in a separate file named myod.m)
% x=x(1);
% y=x(2);
% z=x(3);
dxdydz = zeros(3,1);
dxdydz(1) = x(3) - (1/2)*x(2);
dxdydz(2) = (1/2)*x(1)  - (1/(sqrt(2)))*x(3);
dxdydz(3) = (1/(sqrt(2)))*x(2) - (1/2)*x(1) ;
end