Mathematical application of Jordan canonical form: solving first-order differential equations. The coefficients of the first-order differential equation constitute the matrix A. Usually, the algebraic equation of the eigenvalue of A contains both heteroroots and multiple roots. The similarity transformation of A is generally Jordan block diagonal matrix J (in special cases, it is a pure diagonal matrix), then the exponent of J is Jordan matrix E (JT), and then the standard basic solution matrix E (at) = S E (JT) (S inverse), and E is obtained. Physical application: First-order differential equations are widely used in time-domain dynamic circuits.