# The characteristic equation for the eigenvalues of the Jacobian matrix

The mathematical model is

$$V'=\frac{I}{C}, V(0)=0$$ and $$I'=\frac{-V}{L}-\frac{RI}{L}+\frac{V_β}{L}, I(0)=0$$ With I.C $$V_β=5 V, R=0.1Ω, L=4*10^{-9}H and C=0.5*10^{-9}F$$

I have calculated and I have found that the differential has one equilibrium point $$(V_eq,I_eq)=(5,0)$$.

Now I want to find The characteristic equation for the eigenvalues of the Jacobian matrix.

I use the following code but no result! Where have i done the mistake?

vb = 5; r = 0.1; l = 4*Power[10, -9]; c = 0.5*Power[10, -9];
f = i/c;
g = (-v/l) - (r*i/l) + vb/l;
deq1 = v'[t] == f
deq2 = i'[t] == g
equilibrio = Solve[{f == 0, g == 0}, {v[t], i[t]}]
MatrixForm[jacobian = D[{f, g}, {{v[t], t[t]}}] // Simplify]
Simplify[MatrixForm[jacobian] /. equilibrio]

• Your variables are v and i, not v[t] and i[t]. You should change that in Solve. Dec 26, 2022 at 21:41
• You could see the question mathematica.stackexchange.com/questions/210179/… which shows how to do what you want, I think. Dec 26, 2022 at 22:02

Based on the answer I linked to in the comment above, this may be what you want:

Clear[i, v]

vb = 5; r = 0.1; l = 4*Power[10, -9]; c = 0.5*Power[10, -9];

eqns = {i/c , -v/l - r*i/l + vb/l}

(* {2.*10^9 i, 1250000000 - 2.5*10^7 i - 250000000 v} *)

eq = Solve[eqns == {0, 0}, {v, i}]

(* {{v -> 5., i -> 0.}} *)

eq[]

(* {v -> 5., i -> 0.} *)

j = D[eqns, {{i, v}}]

(* {{2.*10^9, 0}, {-2.5*10^7, -250000000}} *)

j /. eq[]

(* {{2.*10^9, 0}, {-2.5*10^7, -250000000}} *)

CharacteristicPolynomial[j /. eq[], \[Lambda]]

(* -5.*10^17 - 1.75*10^9 \[Lambda] + \[Lambda]^2 *)

Eigenvalues[ j /. eq[] ]

(* {2.*10^9, -2.5*10^8} *)