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Second and Higher Order Linear Differential Equations
Published in Vladimir A. Dobrushkin, Applied Differential Equations, 2022
The Wronskian unambiguously determines whether functions are linearly dependent or independent if and only if these functions are solutions of the same homogeneous differential equation. But what should we do if we don't know whether the functions are solutions of the same differential equation? If these functions are represented as convergent power series, then Theorem 4.8 assures us that evaluating the Wronskian is sufficient. If they are not, there is another advanced tool to determine whether or not a set of functions is linearly independent or dependent, called the Gramm determinant, but it is out of the scope of this book.
Basic Responses of First-Order Circuits
Published in Nassir H. Sabah, Circuit Analysis with PSpice, 2017
The variation of v with time is given by the solution of Equation 11.2. Although the solution can be readily obtained by straightforward integration after separation of variables (Exercise 11.1), let us digress a little and consider a more general approach that will be first applied to the first-order differential equation dydt+yτ=0,t≥0+ where y is a variable that can be a current or a voltage. A differential equation with the variable and its derivatives on the LHS and zero on the RHS, as in Equation 11.3, is referred to as the homogeneous differential equation. The general solution of a homogeneous, linear, ordinary, differential equation of any order, but having constant coefficients, is the sum of exponentials of the form Aest, where A and s are constants that depend on the circuit and on the initial conditions, and where the number of exponential terms in the sum is equal to the order of the differential equation. Since Equation 11.3 is of the first order, its general solution is y(t) = Aest. Substituting in Equation 11.3 and collecting terms, Aest(s+1τ)=0
Sign-changing points of solutions of homogeneous Sturm–Liouville equations with measure-valued coefficients
Published in Applicable Analysis, 2022
For the standard Sturm–Liouville homogeneous differential equation, the Wronskian of two linearly independent solution is a nonzero constant everywhere on which is the key to prove the Sturm separation theorem, see, e.g. Zettl [8]. Unfortunately Equation (11) tells us that the derivative of , even when u and v are linearly independent, is not necessarily the zero measure, unless the two solutions are balanced (i.e. ).