First-order partial differential equation
In mathematics, a first-order partial differential equation is a partial differential equation that involves only first derivatives of the unknown function of "n" variables. The equation takes the form Such equations arise in the construction of characteristic surfaces for hyperbolic partial differential equations, in the calculus of variations, in some geometrical problems, and in simple models for gas dynamics whose solution involves the method of characteristics. If a family of solutions of a single first-order partial differential equation can be found, then additional solutions may be obtained by forming envelopes of solutions in  that family. In a related procedure, general solutions may be obtained by integrating families of ordinary differential equations. General solution and complete integral. 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 is a complete integral if formula_3. The below discussions on the type of integrals are based on the textbook "A Treatise on Differential Equations" (Chaper IX , 6th edition, 1928) by Andrew Forsyth. Complete integral. The solutions are described in relatively simple manner in two or three dimensions with which the key concepts are trivially extended to higher dimensions. A general first-order partial differential equation in three dimensions has the form where formula_5 Suppose formula_6 be the complete integral that contains three arbitrary constants formula_7. From this we can obtain three relations by differentiation formula_8 formula_9 formula_10 Along with the complete integral formula_11, the above three relations can be used to eliminate three constants and obtain an equation (original partial differential equation) relating formula_12. Note that  the elimination of constants leading to the partial differential equation need not be unique, i.e., two different equations can result in the same complete integral, for example, elimination of constants from the relation formula_13 leads to formula_14 and formula_15. General integral. Once a complete integral is found, a general solution can be constructed from it. The general integral is obtained by making the constants functions of the coordinates, i.e., formula_16. These functions are chosen such that the forms of formula_17 are unaltered so that the elimination process from complete integral can be utilized. Differentiation of the complete integral now provides  formula_18 formula_19 formula_20 in which we require the right-hand side terms of all the three equations to vanish identically so that elimination of formula_7 from formula_22 results in the partial differential equation. This requirement can be written more compactly by writing it as formula_23 where is the Jacobian determinant. The condition formula_25 leads to the general solution. Whenever formula_25, then there exists a functional relation between formula_7 because whenever a determinant is zero, the columns (or rows) are not linearly independent. Take this functional relation to be formula_28 Once formula_29 is found, the problem is solved. From the above  relation, we have formula_30. By summing the original equations formula_31, formula_32 and formula_33 we find formula_34. Now eliminating formula_35 from the two equations derived, we obtain formula_36 Since formula_37 and formula_38 are independent, we require formula_39 formula_40 The above two equations can be used to solve formula_37 and formula_38. Substituting formula_7 in formula_11, we obtain the "general integral". Thus a general integral describes a relation between formula_45, two known independent functions formula_29 and an arbitrary function formula_47. Note that we have assumed formula_48 to make the determinant formula_49 zero, but this is not always needed. The relations formula_50 or, formula_51  suffice to make the determinant zero. Singular integral. Singular integral is obtained when formula_52. In this case, elimination of formula_7 from formula_11 works if formula_55 The three equations can be used to solve the three unknowns formula_7. Solution obtained by elimination of formula_7 this way leads to what are called "singular integrals". Special integral. Usually, most integrals fall into three categories defined above, but it may happen that a solution does not fit into any of three types of integrals mentioned above. These solutions are called "special integrals". A relation formula_58 that satisfies the partial differential equation is said to  a "special integral" if we are unable to determine formula_7 from the following equations formula_60 formula_61 formula_62 If we able to determine formula_7 from the above set of equations, then formula_64 will turn out to be one of the three integrals described before. Two dimensional case. The complete integral in two-dimensional space can be written as formula_65. The general integral is obtained by eliminating formula_37 from the following equations formula_67 The singular integral if it exists can be obtained by eliminating formula_29 from the following equations formula_69 If a complete integral is not available, solutions may still be obtained  by solving a system of ordinary equations. To obtain this system, first note that the PDE determines a cone (analogous to the light cone) at each point: if the PDE is linear in the derivatives of "u" (it is quasi-linear), then the cone degenerates into a line. In the general case, the pairs ("p","q") that satisfy the equation determine a family of planes at a given point: where The envelope of these planes is a cone, or a line if the PDE is quasi-linear. The condition for an envelope is where F is evaluated at formula_73, and "dp  and dq" are increments of "p" and "q" that satisfy "F"=0. Hence the generator of the cone is a line with direction This direction corresponds to the light rays for the wave equation. To integrate differential equations along these directions, we require increments for "p" and "q" along the ray. This can be obtained by differentiating the PDE: Therefore the ray direction in formula_77 space is The integration of these equations leads to a ray conoid at each point formula_79. General solutions of the PDE can then be obtained from envelopes of such conoids. Definitions of linear dependence for differential  systems. This part can be referred to formula_80 of Courant's book. An equivalent description is given. Two definitions of linear dependence are given for first-order linear partial differential equations. Where formula_82 are independent variables; formula_83 are dependent unknowns; formula_84 are linear coefficients; and formula_85 are non-homogeneous items. Let formula_86. Definition I: Given a number field formula_87, when there are coefficients (formula_88), not all zero, such that formula_89, this definition degenerates into the definition I. The div-curl systems, Maxwell's equations, Einstein's equations (with four harmonic coordinates) and Yang-Mills equations (with gauge conditions) are well -determined in definition II, whereas are over-determined in definition I. Characteristic surfaces for the wave equation. Characteristic surfaces for the wave equation are level surfaces for solutions of the equation There is little loss of generality if we set formula_91: in that case "u" satisfies In vector notation, let A family of solutions with planes as level surfaces is given by where If "x" and "x" are held fixed, the envelope of these solutions is obtained by finding a point on the sphere of radius 1/"c" where the value of "u" is stationary. This is true if formula_96 is  parallel to formula_97. Hence the envelope has equation These solutions correspond to spheres whose radius grows or shrinks with velocity "c". These are light cones in space-time. The initial value problem for this equation consists in specifying a level surface "S" where "u"=0 for "t"=0. The solution is obtained by taking the envelope of all the spheres with centers on "S", whose radii grow with velocity "c". This envelope is obtained by requiring that This condition will be satisfied if formula_100 is normal to "S". Thus the envelope corresponds to motion with velocity "c" along each normal to "S . This is the Huygens' construction of wave fronts: each point on S" emits a spherical wave at time "t"=0, and the wave front at a later time "t" is the envelope of these spherical waves. The normals to "S" are the light rays.