Liouville's formula

In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of the same mathematical relationship. c 1 := det Φ ( x ) = x y 1 ( x ) − y 2 ( x ) , x ∈ I , {displaystyle c_{1}:=det Phi (x)=x,y_{1}(x)-y_{2}(x),qquad xin I,}     (1) ( det Φ ) ′ = ∑ i = 1 n det ( Φ 1 , 1 Φ 1 , 2 ⋯ Φ 1 , n ⋮ ⋮ ⋮ Φ i , 1 ′ Φ i , 2 ′ ⋯ Φ i , n ′ ⋮ ⋮ ⋮ Φ n , 1 Φ n , 2 ⋯ Φ n , n ) . {displaystyle (det Phi )'=sum _{i=1}^{n}det {egin{pmatrix}Phi _{1,1}&Phi _{1,2}&cdots &Phi _{1,n}\vdots &vdots &&vdots \Phi '_{i,1}&Phi '_{i,2}&cdots &Phi '_{i,n}\vdots &vdots &&vdots \Phi _{n,1}&Phi _{n,2}&cdots &Phi _{n,n}end{pmatrix}}.}     (2) ( det Φ ) ′ = ∑ i = 1 n a i , i det Φ = t r A det Φ {displaystyle (det Phi )'=sum _{i=1}^{n}a_{i,i}det Phi =mathrm {tr} ,A,det Phi }     (3) In mathematics, Liouville's formula, also known as the Abel-Jacobi-Liouville Identity, is an equation that expresses the determinant of a square-matrix solution of a first-order system of homogeneous linear differential equations in terms of the sum of the diagonal coefficients of the system. The formula is named after the French mathematician Joseph Liouville. Jacobi's formula provides another representation of the same mathematical relationship. Liouville's formula is a generalization of Abel's identity and can be used to prove it. Since Liouville's formula relates the different linearly independent solutions of the system of differential equations, it can help to find one solution from the other(s), see the example application below. Consider the n-dimensional first-order homogeneous linear differential equation on an interval I of the real line, where A(x) for x ∈ I denotes a square matrix of dimension n with real or complex entries. Let Φ denote a matrix-valued solution on I, meaning that each Φ(x) is a square matrix of dimension n with real or complex entries and the derivative satisfies