Substituting a function from a differential equation

Hi, I am a beginner and I am learning symbolic math toolbox for MATLAB.

I have got a system of differential equations

and a definition for alpha:

where r = √(x^2 +y^2 +z^2).

My aim is to differentiate alpha with respect to t, and then substitute the differential equations to obtain

My code looks like this:

syms alpha(t) beta(t) r(t) x(t) y(t) z(t) t epsilon theta(t)
r(t) = sqrt((x(t))^2+(y(t))^2+(z(t))^2)
dtheta(t)=diff(theta(t),t)
alpha(t) = sqrt(( r+z(t))/(2*r))
beta(t) = (x(t)+ i*y(t))/sqrt(2*r*(r+z(t))) %I want to do the same for beta too, differentiate it w.r.t t and substitute the differential equations
diffeqn1 = epsilon*diff(x(t),t) == -y(t) - epsilon*dtheta(t)*z(t)
diffeqn2 = epsilon*diff(y(t),t) == x(t)
diffeqn3 = diff(z(t),t) == dtheta(t) *  x(t)

diff(alpha(t),t)

But of course, I only get an expression involving the derivatives of x(t),y(t),z(t).

How can I make the program take diffeqn1 ,diffeqn2, diffeqn3 into consideration?

Any help would be massively appreciated.

[Sean Thrasher]

I am using MATLAB_R2020a

Answer 1

[Surya Talluri]

I understand that you want to substitute diff(x(t),t), diff(x(t),t), diff(x(t),t) values in diff(alpha(t),t). You can use “isolate” function to obtain differentiation equations and substitute them in diff(alpha(t),t).

dx = diff(x(t),t); 
dy = diff(y(t),t); 
dz = diff(z(t),t); 
 
diffeqn1 = epsilon*dx == -y(t) - epsilon*dtheta(t)*z(t); 
diffeqn2 = epsilon*dy == x(t); 
diffeqn3 = dz == dtheta(t) *  x(t); 
 
diffeqn1 = isolate(diffeqn1, dx); 
diffeqn2 = isolate(diffeqn2, dy); 
diffeqn3 = isolate(diffeqn3, dz); 
 
dalpha = diff(alpha(t),t); 
 
dalpha = subs(dalpha, {lhs(diffeqn1), lhs(diffeqn2), lhs(diffeqn3)}, {rhs(diffeqn1), rhs(diffeqn2), rhs(diffeqn3)}); 

You can refer to the following documentation for further understanding:

• isolate - https://www.mathworks.com/help/symbolic/isolate.html
• lhs - https://www.mathworks.com/help/symbolic/lhs.html
• rhs - https://www.mathworks.com/help/symbolic/rhs.html
• subs - https://www.mathworks.com/help/symbolic/subs.html