Solution by differentiation

Case i. The equation can he split up into factors. If the differential equation can be resolved into n factors of the first degree, equate each factor to zero and solve each of the n equations separately. The n solutions may be left either distinct, or combined into one. 

Examples.—(1) Solve x(dyjdx)2=y. Resolve into factors of the first degree, dxjdyss + tjyjx. Separate the variables and integrate, x dx y dy— + /C, where sJC is the integration constant. Hence Jx sjy = + JC, which, on rationalization, becomes (x - y)2 - 2C x + y) + C2 = 0. Geometrically this equation represents a system of parabolic curves each of which touches the axis at a distance G from the origin. The separate equations of the above solution merely represent different branches of the same parabola. 

Case ii. The equation cannot he resolved into factors, hut it can he solved for x, y, dyjdx, or y/x. An equation which cannot be resolved into factors, can often be expressed in terms of x, y, dyjdx, or yjx, according to circumstances. The differential coefficient of the one variable with respect to the other may be then obtained by solving for dyjdx and using the result to eliminate dyjdx from the given equation. 

Separate the variables x andp, solve for dyjdx, and integrate by the method of partial fractions. 

Case iii. The equation cannot be resolved into factors, x or y is absent. If x is absent solve for dyjdx or y according to convenience if y is absent, solve for dxjdy or x. Differentiate the result with respect to the absent letter if necessary and solve in the regular way. 

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Solution By differentiating >ab(M0) (Example 3.19) with respect to X, we obtain... 

Rsn = gas remaining in solution by differential vaporization, page 5, column 2, Table 10-1... 

Her LM, Nail SL. Measurement of glass transition temperatures of freezeconcentrated solutes by differential scanning calorimetry. Pharm Res 1994 11 54-59. 

Similar arguments and definitions can be applied to the other partial molar thermodynamic functions and properties of the components in solution. By differentiation of Equation (8.71), the following expressions for the partial molar entropy, enthalpy, volume, and heat capacity of the kth component are obtained ... 

Hao, Z., Iqbal, A. and Herren, F. (1999c). l,4-Diketo-3,6-bis-4-chloro-phenyl-pyrrolopyrrole in /3 and y modification, prepd. by acid hydrolysis of soluble carbamate. Derivation and preparation by cooling. Ciba-Geigy Ltd. EP 690059 B1. [271 ] Harano, Y. and Oota, K. (1978). Measurement of crystallization of potassium bro-mate from its quiescent aqueous solution by differential scanning calorimeter. Homogeneous nucleation rate. /. Chem. Eng. J., 11, 159-61. [70]... 

L. Gatlin and P. Deluca, A study of the phase separation in frozen antibiotic solutions by differential scanning calorimetry. J. Parent. Drug Assoc. 34 398—408 (1980). 

In this method salicylic acid is determined in its compound preparations. A sample of 1 g of ointment or 1 ml of liquid preparation, containing 6% of salicylic acid and 12% of benzoic acid is dissolved in boiling water. The cooled solution is diluted to 500 ml with H20 and filtered, and a 100 ml portion of the titrate is diluted to 100 ml with H20. Salicylic acid is determined in the resulting solution by differential spectrophotometry at 297 nm (22). 

Antonsen, K. P., Hoffman, A. S., Water structure of PEG solutions by differential scanning calorimetry measurements, in Poly(ethylene glycol) Chemistry Biotechnical and Biomedical applications (J.M. Harris, Ed.), Plenum Press, New York, 1992, pp. 15-28. 

Differential Equations of the First Order and of the First or Higher Degree.—Solution by Differentiation. 

The cloud point, usually between 0 and -10°C, is determined visually (as in NF T 07-105). It is equal to the temperature at which paraffin crystals normally dissolved in the solution of all other components, begin to separate and affect the product clarity. The cloud point can be determined more accurately by differential calorimetry since crystal formation is an exothermic phenomenon, but as of 1993 the methods had not been standardized. 

The finite element solution of differential equations requires function integration over element domains. Evaluation of integrals over elemental domains by analytical methods can be tedious and impractical and is not attempted in... 

The simplicity gained by choosing identical weight and shape functions has made the standard Galerkin method the most widely used technique in the finite element solution of differential equations. Because of the centrality of this technique in the development of practical schemes for polymer flow problems, the entire procedure of the Galerkin finite element solution of a field problem is further elucidated in the following worked example. 

The concentration of As(III) in water can be determined by differential pulse polarography in 1 M HCl. The initial potential is set to -0.1 V versus the SCE, and is scanned toward more negative potentials at a rate of 5 mV/s. Reduction of As(III) to As(0) occurs at a potential of approximately —0.44 V versus the SCE. The peak currents, corrected for the residual current, for a set of standard solutions are shown in the following table. 

The amount of sulfur in aromatic monomers can be determined by differential pulse polarography. Standard solutions are prepared for analysis by dissolving 1.000 mb of the purified monomer in 25.00 mb of an electrolytic solvent, adding a known amount of S, deaerating, and measuring the peak current. The following results were obtained for a set of calibration standards... 

We verify the solution by carrying out the indicated differentiation and substituting back into the differential equation ... 

Fluid deposits are defined as those which can be recovered in fluid form by pumping, in solution, or as particles in a slurry. Petroleum products and Frasch process sulfur are special cases. At this time no vaUd distinction is made between resources on the continental shelf and in the deep oceans. However, deep seabed deposits of minerals which can be separated by differential solution are expected to be amenable to fluid mining methods in either environment. 

Quenching. After solution treatment, the product is generally cooled to room temperature at such a rate to retain essentially all of the solute in solution. The central portions of thicker products caimot be cooled at a sufficient rate to prevent extensive precipitation in some alloys. Moreover, some forgings and castings are dehberately cooled slowly to minimize distortion and residual stress produced by differential cooling in different portions of the products. Cold water, either by immersion or by sprays, is the most commonly used cooling medium. Hot water or a solution of a polymer in cold water is used when the highest rates are not desired. Dilute Al—Mg—Si and Al—Mg—Zn extmsions can be effectively solution heat treated by the extmsion process therefore, they may be quenched at the extmsion press by either air or water. 

Integral-Transform Method A number of integral transforms are used in the solution of differential equations. Only one, the Laplace transform, will be discussed here [for others, see Integral Transforms (Operational Methods) ]. The one-sided Laplace transform indicated by L[f t)] is defined by the equation L[f t)] = /(O dt. It has... 

Optimal design of ammonia synthesis by differential equation solution and a numerical gradient search... 

The development of mathemafical models is described in several of the general references [Giiiochon et al., Rhee et al., Riithven, Riithven et al., Suzuki, Tien, Wankat, and Yang]. See also Finlayson [Numerical Methods for Problems with Moving Front.s, Ravenna Park, Washington, 1992 Holland and Liapis, Computer Methods for Solving Dynamic Separation Problems, McGraw-Hill, New York, 1982 Villadsen and Michelsen, Solution of Differential Equation Models by... 

Because the solution is electrically neutral, conservation of charge is expressed by differentiating Eq. (22-16) ... 

Differential equations and their solutions will be stated for the elementary models with the main lands of inputs. Since the ODEs are linear, solutions by Laplace transforms are feasible. 

Once the elution-curve equation is derived, and the nature of f(v) identified, then by differentiating f(v) and equating to zero, the position of the peak maximum can be determined and an expression for the retention volume (Vr) obtained. The expression for (Vr) will disclose those factors that control solute retention. 

The retention volume of a solute is that volume of mobile phase that passes through the column between the injection point and the peak maximum. Consequently, by differentiating equation (10), equating to zero and solving for (v), an expression for the retention volume (Vr) can be obtained. 

It is seen that all the points lie on the same straight line, irrespective of the operating temperature and, thus, the enthalpy term is close to zero and the solutes are not retained by differential molecular forces. Thus, the curve shows the effect of... 

The equation for the retention volume of a solute, that was derived by differentiating the elution curve equation, can be used to obtain an equation for the retention time of a solute (tr) by dividing by the flowrate (Q), thus,... 

Now equation (39) is a standard integral, the solution of which can be seen by differentiation to be... 

The validity of this solution can be confirmed by differentiation as follows ... 

An accurate indication is achieved by carrying out the calculations in small time steps, such as At = 0.004 s, and then by calculating the vaporization, humidity change, and corresponding temperature rise at each time step. This is the numerical solution of differential equations (4.326) and (4.328). The results of a calculation of this type are shown in Table 4.12. 

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