Linear first-order differential equations solution

This is a linear, first-order differential equation, the solution of which is... 

Use the integrating factor method to find the general solutions to first-order differential equations linear in y... 

Linear Equations A differential equation is said to be linear when it is of first degree in the dependent variable and its derivatives. The general linear first-order differential equation has the form dy/dx + P x)y = Q x). Its general solution is... 

Equations (4-21) are linear first-order differential equations. We considered in detail the solution of such sets of rate equations in Section 3-2, so it is unnecessary to carry out the solutions here. In relaxation kinetics these equations are always solved by means of the secular equation, but the Laplace transformation can also be used. Let us write Eqs. (4-21) as... 

It should be observed that k.3 was approximated to zero in the above treatment. The differential equations describing B and C concentrations are linear ones with respect to the participating concentrations. The expression for the A-concentration is inserted in eqs. (10b) and (10c) and the first order differential equations are solved with the initial conditions Cb=0 and Cc=0 at t=0. The solutions become... 

For each occupied orbital y = 1,..., n, we have to solve the set of N -h 1 linear first-order differential equations (21). Unfortunately, it does not seem possible to obtain analytical solutions for any realistic choice of F(t) and, therefore, it is necessary to resort to finding approximate or numerical... 

The mathematical problem posed is the solution of the simultaneous differential equations which arise from the mass-action treatment of the chemistry. For the homogeneous, well-mixed reactor, this becomes a set of ordinary, non-linear, first-order differential equations. For systems that are not... 

Unfortunately, unlike the general linear first-order differential equation (7.31), there is no simple template which provides the solution, and we need therefore to apply different methods to suit the equation we meet in the chemical context. Equations of the general form given in equation (7.45) crop up in all branches of the physical sciences where a system is under the influence of an oscillatory or periodic change. In chemistry, some of the most important examples can be found in modelling ... 

Finding general solutions to linear first order differential equations using the integrating factor method. 

The radiation balance of a layer with the thickness d having an infinitely large surface, irradiated homogeneously from one side with exciting radiation, is given by the solution of four coupled linear first-order differential equations (Eqs. 3.5-1...4). This is a boundary value problem, with the definitions given in Fig. 3.5-2. We are discussing... 

Solution of Eqns. (37-40) can be obtained by simultaneous integration of the resulting 2(Nq+N [) coupled linear first-order differential equations by difference techniques as presented in Section V. For time-independent Hamiltonians H the total energy... 

Example 2.1 Linear First-Order Differential Equation In the development of boundary-layer mass transfer to a planar electrode, a similarity transformation variable (see Section 2.4) can be identified through solution of... 

Remember 2.1 The general solution to nonhomogeneous linear first-order differential equations can be obtained as the product of function to be determined and the solution to the homogeneous equation. 

The general solution (3-6) to a set of linear first order differential equations such as Eqs. (5) is well known it is... 

This linear first-order differential equation has the solution... 

Since Eq. (2-104) is a linear first-order differential equation, it has an analytic solution. With the stated initial condition the result can be expressed in terms of the yield of... 

Thus we are confronted with the solution of a two-point boundary value problem for a linear system of first order differential equations. In the event that the rod is inhomogeneous, the matrices A, , (7, or D will depend on y. 

The direct problem of chemical kinetics always has an analytical solution if a reaction mathematical model is a linear system of ordinary first-order differential equations. Sequences of elementary first-order kinetic steps, including ones complicated with reversible and competitive steps, correspond to such mathematical models. Let us mark off the classical matrix method firom analytical methods of solving such ODE systems. 

The simplest situation is found when the bulk concentrations are kept constant by an appropriate stirring device (usually RDE), hence AC7 = 0. The AC are given by the solution of the set of ordinary linear first-order differential equations obtained by linearization of Eq. (2) under a sine wave potential perturbation IsE = A exp jtat. Resolution by the Kramers method immediately shows that AC /AE is expressed by a rational function of the imaginary angular frequency jar. 

Often, Laplace transforms are used to find solutions to linear first-order differential equations, particularly when a set of linear firsf order differential... 

The ability to solve nonlinear differential equations as readily as linear equations is one of the major advantages of the numerical solution of differential equations. For one such example, the Van der Pol equation is a classical nonlinear equation that has been extensively studied in the literature. It is defined in second order form and first order differential equation form as ... 

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