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Let ˙ = be a linear first order differential equation, where () is a column vector of length and () an periodic matrix with period (that is (+) = for all real values of ). Let ϕ ( t ) {\displaystyle \phi \,(t)} be a fundamental matrix solution of this differential equation.
For a first-order PDE, the method of characteristics discovers so called characteristic curves along which the PDE becomes an ODE. [1] [2] Once the ODE is found, it can be solved along the characteristic curves and transformed into a solution for the original PDE.
For an arbitrary system of ODEs, a set of solutions (), …, are said to be linearly-independent if: + … + = is satisfied only for = … = =.A second-order differential equation ¨ = (,, ˙) may be converted into a system of first order linear differential equations by defining = ˙, which gives us the first-order system:
First-order means that only the first derivative of y appears in the equation, and higher derivatives are absent. Without loss of generality to higher-order systems, we restrict ourselves to first-order differential equations, because a higher-order ODE can be converted into a larger system of first-order equations by introducing extra variables.
Ince, E. L. (1956), Ordinary differential equations, New York: Dover Publications, ISBN 0486603490. (This is a classic reference on ODEs, first published in 1926.) Andrei D. Polyanin; Valentin F. Zaitsev (15 November 2017). Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems. CRC Press. ISBN 978-1-4665-6940-9.
The coefficients of the super-harmonic terms are solved directly, and the coefficients of the harmonic term are determined by expanding down to order-(n+1), and eliminating its secular term. See chapter 10 of [5] for a derivation up to order 3, and [8] for a computer derivation up to order 164.
The original problem was stated as follows (English translation from 1902): Proof of the existence of linear differential equations having a prescribed monodromic group In the theory of linear differential equations with one independent variable z, I wish to indicate an important problem one which very likely Riemann himself may have had in mind.
The general solution to the first order partial differential equation is a solution which contains an arbitrary function. But, the solution to the first order partial differential equations with as many arbitrary constants as the number of independent variables is called the complete integral. The following n-parameter family of solutions