Solve symbolic equation by comparing coefficients

I have an equation like this: , which I need to rearrange to this form: .

Obviously, the above holds true for and . However, I would like to do this automatically. How can d and e be found using Matlabs Symbolic Toolbox?

The following approach won't help, since the system isn't well defined.

syms a b c d e f x y z

eq = a*x + y*(b + c*z)==0;

target = x + y*(d + e*z)==0;

solve([eq target],[d e])

Any help would be appreciated!

[Highphi]

syms a b c d e f x y z
eq(a,b,c) = a*x + y*(b + c*z);
target(a,b,c,d,e) = x + y*(d + e*z);

eq2(a,b,c,d,e) = subs(eq, [b, c], [d*a, e*a])

eq2(a, b, c, d, e) = 

simplify(eq2)

ans(a, b, c, d, e) = 

D = solve(eq2 == 0, d)

D = 

E = solve(eq2 == 0, e)

E = 

not sure if that helps much but maybe it'll help you get closer to where you want to be. (I'm not totally positive what you're trying to do with it 😝 )

[Jörn Froböse]

Hey Highphi, thank you for your suggestion. This would'nt be really helpful in my case, since I need coefficients d and e without any dependence on x, y and z. I kind of solved it now by "brute-forcing" and using subs.

Answer 1

[Prachi Kulkarni]

Hi,

For a given example expression of the form a*x + y*(b + c*x), you can get the coefficients d and e of the reduced forms as follows.

syms x y z
eq = 2*x + y*(3 + 4*z); % input equation
[d,e] = reducedCoefficients(eq);

function [d,e] = reducedCoefficients(eq)
    coef = coeffs(eq);
    b = coef(1);
    c = coef(2);
    a = coef(3);
    
    d = double(b/a);
    e = double(c/a);
end

[Jörn Froböse]

Thank you for your answer! This isn't really helpful in my special case, since a, b, c are symbolic coefficients.

[Jörn Froböse]

More specifically in my case, the problem is to convert an equation of, lets say, this form:

into this form (Mathieu function):

with , . How can λ and γ be found generally?

[Prachi Kulkarni]

Hi,

For the general case, you can get the coefficients d and e of the reduced form as follows.

syms a b c x y z
eq = a*x + y*(b + c*z);

coef = coeffs(eq,[x,y,z]);
b = coef(1);
c = coef(2);
a = coef(3);

d = b/a;
e = c/a;

For your specific case, you can get the coefficients lambda and gamma of the reduced form as follows.

syms eta a Q d2ydt2 y t
eq = 4*(eta.^2)*d2ydt2 + 1.5*a*(Q.^2)*(1+cos(t))*y + y;

coef = coeffs(eq,[d2ydt2,y,cos(t)]);
lambda = coef(1)/coef(3);
gamma = coef(2)/coef(3);

[Jörn Froböse]

Thank you, this works for me!