Sentence examples for a formula for the solution from inspiring English sources

There is one special case for which we can give a formula for the solution (Q_{X,Y}) to (3.5): When X and Y commute, (Q_{X,Y} = XY^{-1}).

First we obtain a formula for the solution of the following initial value problem for a scalar linear impulsive differential equation with random moments of impulses and random amplitude of jumps: u ′ = a ( t ) u for  t ≥ 0, ξ k < t < ξ k + 1, u ( ξ k + 0 ) = B k ( τ k ) u ( ξ k − 0 ) for  k = 1, 2, …, u ( t 0 ) = u 0, (10).

So, we derive a formula for the general solution.

In the previous section, we derived a formula of the solution (see Theorem 4.1 for ) (5.17).

The main purpose will be to provide necessary and sufficient conditions for the existence and uniqueness of solutions of the above boundary value problem, i.e., to study the conditions under which the system has unique, infinite and no solutions and to provide a formula for the case of the unique solution (if it exists).

Our above endeavors were to develop a formula for the calculation of exact solutions for the velocity field and the shear stress of the motion (flow) of an Oldroyd-B fluid present between two rotationally oscillating cylinders of infinite lengths.

Furthermore, we provide a formula for the case of the unique solution.

(19) Requiring non-trivial solutions gives a formula for the spectrum as (det[mathcal{A} - lambda+1) I]=0), which yields lambda_{pm}= -1 + frac{operatorname{Tr} mathcal{A} pmsqrt { operatorname{Tr} mathcal{A})^{2} - 4 detmathcal{A}}}{2}.

The existence of a solution to the inverse source problem is shown in appropriate function spaces and a representation formula for the solution is proposed.

These two properties enable us to construct a semi-analytic formula for the solution associated with piecewise affine initial, boundary and internal conditions.

Employing such obtained formulas for (a_{k}) and (y_{k}) in (25), we obtain a closed form formula for the solution to (15).