Handbook of mathematics for engineers and scienteists part 185 pptx

1256 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 9. ∂w ∂x +  xf(w) + yg(w) + h(w)  ∂w ∂y =0. General solution: y + xf(w)+h(w) g(w) + f(w) g 2 (w) =exp  g(w)x  Φ(w). 10. ∂w ∂x + f(x)g(y)h(w) ∂w ∂y =0. General solution:  dy g(y) – h(w)  f(x) dx = Φ(w). T7.2.3. Equations of the Form ∂w ∂x + f(x, y, w) ∂w ∂y = g(x, y, w)  In the solutions of equations T7.2.3.1–T7.2.3.11, Φ(z) is an arbitrary composite function whose argument z can depend on x, y, and w. 1. ∂w ∂x + aw ∂w ∂y = f (x). General solution: y = ax  w – F(x)  + a  F (x) dx + Φ  w – F(x)  ,whereF (x)=  f(x) dx. 2. ∂w ∂x + aw ∂w ∂y = f (y). General solution: x =  y y 0 dz √ 2aF (z)–2au + Φ(u), where F (y)=  f(y) dy, u = F (y)– 1 2 aw 2 . 3. ∂w ∂x +  aw + f(x)  ∂w ∂y = g(x). General solution: y = ax  w – G(x)  + a  G(x) dx + F(x)+Φ  w – G(x)  , where F (x)=  f(x) dx, G(x)=  g(x) dx. 4. ∂w ∂x + f(w) ∂w ∂y = g(x). General solution: y =  x x 0 f  G(t)–G(x)+w  dt +Φ  w – G(x)  ,whereG(x)=  g(x) dx. T7.2. QUASILINEAR EQUATIONS 1257 5. ∂w ∂x + f(w) ∂w ∂y = g(y). General solution: x =  y y 0 ψ  G(t)–G(y)+F (w)  dt + Φ  F (w)–G(y)  , where G(y)=  g(y) dy and F (w)=  f(w) dw. The function ψ = ψ(z)isdefined parametrically by ψ = 1 f(w) , z = F (w). 6. ∂w ∂x + f(w) ∂w ∂y = g(w). General solution: y =  f(w) g(w) dw + Φ  x –  dw g(w)  . 7. ∂w ∂x +  f(w) + g(x)  ∂w ∂y = h(x). General solution: y =  x x 0 f  H(t)–H(x)+w  dt + G(x)+Φ  w – H(x)  , where G(x)=  g(x) dx, H(x)=  h(x) dx. 8. ∂w ∂x +  f(w) + g(x)  ∂w ∂y = h(w). General solution: y =  f(w) h(w) dw +  w w 0 g  H(t)–H(w)+x  h(t) dt + Φ  x – H(w)  ,whereH(x)=  dw h(w) . 9. ∂w ∂x +  f(w) + yg(x)  ∂w ∂y = h(x). General solution: yG(x)–  G(x)f  H(t)–H(x)+w  dx = Φ  w – H(x)  , where G(x)=exp  –  g(x) dx  and H(x)=  h(x) dx. 10. ∂w ∂x + f(x, w) ∂w ∂y = g(x). General solution: y =  x x 0 f  t, G(t)–G(x)+w  dt+Φ  w–G(x)  ,whereG(x)=  g(x) dx. 1258 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 11. ∂w ∂x + f(x, w) ∂w ∂y = g(w). General solution: y =  w w 0 f  G(t)–G(w)+x, t  g(t) dt+Φ  x–G(w)  ,whereG(w)=  dw g(w) . T7.3. Nonlinear Equations T7.3.1. Equations Quadratic in One Derivative  In this subsection, only complete integrals are presented. In order to construct the corresponding general solution, one should use the formulas of Subsection 13.2.1. 1. ∂w ∂x + a  ∂w ∂y  2 = by. This equation governs the free vertical drop of a point body near the Earth’s surface (y is the vertical coordinate measured downward, x time, m = 1 2a the mass of the body, and g = 2ab the gravitational acceleration). Complete integral: w =–C 1 x 2a 3b  by + C 1 a  3/2 + C 2 . 2. ∂w ∂x + a  ∂w ∂y  2 + by 2 =0. This equation governs free oscillations of a point body of mass m = 1/(2a) in an elastic field with elastic coefficient k = 2b (x is time and y is the displacement from the equilibrium). Complete integral: w =–C 1 x + C 2   C 1 – by 2 a dx + C 2 . 3. ∂w ∂x + a  ∂w ∂y  2 = f (x) + g(y). Complete integral: w =–C 1 x +  f(x) dx +   g(y)+C 1 a dy + C 2 . 4. ∂w ∂x + a  ∂w ∂y  2 = f (x)y + g(x). Complete integral: w = ϕ(x)y +   g(x)–aϕ 2 (x)  dx + C 1 ,whereϕ(x)=  f(x) dx + C 2 . 5. ∂w ∂x + a  ∂w ∂y  2 = f (x)w + g(x). Complete integral: w = F (x)(C 1 + C 2 y)+F (x)   g(x)–aC 2 2 F 2 (x)  dx F (x) ,whereF(x)=exp   f(x) dx  . T7.3. NONLINEAR EQUATIONS 1259 6. ∂w ∂x – f(w)  ∂w ∂y  2 =0. Complete integral in implicit form:  f(w) dw = C 2 1 x + C 1 y + C 2 . 7. f 1 (x) ∂w ∂x + f 2 (y)  ∂w ∂y  2 = g 1 (x) + g 2 (y). Complete integral: w =  g 1 (x)–C 1 f 1 (x) dx +   g 2 (y)+C 1 f 2 (y) dy + C 2 . 8. ∂w ∂x + a  ∂w ∂y  2 + b ∂w ∂y = f (x) + g(y). Complete integral: w =–C 1 x + C 2 +  f(x) dx – b 2a y 1 2a   4ag(y)+b 2 + 4aC 1 dy. 9. ∂w ∂x + a  ∂w ∂y  2 + b ∂w ∂y = f (x)y + g(x). Complete integral: w = ϕ(x)y +   g(x)–aϕ 2 (x)–bϕ(x)  dx + C 1 ,whereϕ(x)=  f(x) dx + C 2 . 10. ∂w ∂x + a  ∂w ∂y  2 + b ∂w ∂y = f (x)w + g(x). Complete integral: w =(C 1 y+C 2 )F (x)+F (x)   g(x)–aC 2 1 F 2 (x)–bC 1 F (x)  dx F (x) , F(x)=exp   f(x) dx  . T7.3.2. Equations Quadratic in Two Derivatives 1. a  ∂w ∂x  2 + b  ∂w ∂y  2 = c. For a = b,thisisadifferential equation of light rays. Complete integral: w = C 1 x + C 2 y + C 3 ,whereaC 2 1 + bC 2 2 = c. An alternative form of the complete integral: w 2 c = (x – C 1 ) 2 a + (y – C 2 ) 2 b . 2.  ∂w ∂x  2 +  ∂w ∂y  2 = a –2by. This equation governs parabolic motion of a point mass in vacuum (the coordinate x is measured along the Earth’s surface, the coordinate y is measured vertically upward from the Earth’s surface, and a is the gravitational acceleration). Complete integral: w = C 1 x 1 3b (a – C 2 1 – 2by) 3/2 + C 2 . 1260 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 3.  ∂w ∂x  2 +  ∂w ∂y  2 = a  x 2 + y 2 + b. This equation arises from the solution of the two-body problem in celestial mechanics. Complete integral: w =   b + a r – C 2 1 r 2 dr + C 1 arctan y x + C 2 ,wherer =  x 2 + y 2 . 4.  ∂w ∂x  2 +  ∂w ∂y  2 = f (x). Complete integral: w = C 1 y + C 2   f(x)–C 2 1 dx. 5.  ∂w ∂x  2 +  ∂w ∂y  2 = f (x) + g(y). Complete integral: w =   f(x)+C 1 dx   g 2 (y)–C 1 dy + C 2 . The signs before each of the integrals can be chosen independently of each other. 6.  ∂w ∂x  2 +  ∂w ∂y  2 = f (x 2 + y 2 ). Hamilton’s equation for the plane motion of a point mass under the action of a central force. Complete integral: w = C 1 arctan x y + C 2 1 2   zf(z)–C 2 1 dz z , z = x 2 + y 2 . 7.  ∂w ∂x  2 +  ∂w ∂y  2 = f (w). Complete integral in implicit form:  dw  f(w) =  (x + C 1 ) 2 +(y + C 2 ) 2 . 8.  ∂w ∂x  2 + 1 x 2  ∂w ∂y  2 = f (x). This equation governs the plane motion of a point mass in a central force field, with x and y being polar coordinates. Complete integral: w = C 1 y   f(x)– C 2 1 x 2 dx + C 2 . 9.  ∂w ∂x  2 + f(x)  ∂w ∂y  2 = g(x). Complete integral: w = C 1 y + C 2 +   g(x)–C 2 1 f(x) dx. T7.3. NONLINEAR EQUATIONS 1261 10.  ∂w ∂x  2 + f(y)  ∂w ∂y  2 = g(y). Complete integral: w = C 1 x + C 2 +   g(y)–C 2 1 f(y) dy. 11.  ∂w ∂x  2 + f(w)  ∂w ∂y  2 = g(w). Complete integral in implicit form:   C 2 1 + C 2 2 f(w) g(w) dw = C 1 x + C 2 y + C 3 . One of the constants C 1 or C 2 can be set equal to 1. 12. f 1 (x)  ∂w ∂x  2 + f 2 (y)  ∂w ∂y  2 = g 1 (x) + g 2 (y). A separable equation. This equation is encountered in differential geometry in studying geodesic lines of Liouville surfaces. Complete integral: w =   g 1 (x)+C 1 f 1 (x) dx   g 2 (y)–C 1 f 2 (y) dy + C 2 . The signs before each of the integrals can be chosen independently of each other. T7.3.3. Equations with Arbitrary Nonlinearities in Derivatives 1. ∂w ∂x + f  ∂w ∂y  =0. This equation is encountered in optimal control and differential games. 1 ◦ . Complete integral: w = C 1 y – f(C 1 )x + C 2 . 2 ◦ . On differentiating the equation with respect to y, we arrive at a quasilinear equation of the form T7.2.2.3: ∂u ∂x + f  (u) ∂u ∂y = 0, u = ∂w ∂y , which is discussed in detail in Subsection 13.1.3. 3 ◦ . The solution of the Cauchy problem with the initial condition w(0, y)=ϕ(y) can be written in parametric form as y = f  (ζ)x + ξ, w =  ζf  (ζ)–f (ζ)  x + ϕ(ξ), where ζ = ϕ  (ξ). See also Examples 1 and 2 in Subsection 13.2.3. 2. ∂w ∂x + f  ∂w ∂y  = g(x). Complete integral: w = C 1 y – f(C 1 )x +  g(x) dx + C 2 . 1262 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 3. ∂w ∂x + f  ∂w ∂y  = g(x)y + h(x). Complete integral: w = ϕ(x)y +   h(x)–f  ϕ(x)  dx + C 1 ,whereϕ(x)=  g(x) dx + C 2 . 4. ∂w ∂x + f  ∂w ∂y  = g(x)w + h(x). Complete integral: w =(C 1 y + C 2 )ϕ(x)+ϕ(x)   h(x)–f(C 1 ϕ(x))  dx ϕ(x) ,whereϕ(x)=exp   g(x) dx  . 5. ∂w ∂x – F  x, ∂w ∂y  =0. Complete integral: w =  F (x, C 1 ) dx + C 1 y + C 2 . 6. ∂w ∂x + F  x, ∂w ∂y  = aw. Complete integral: w = e ax (C 1 y + C 2 )–e ax  e –ax F (x, C 1 e ax ) dx. 7. ∂w ∂x + F  x, ∂w ∂y  = g(x)w. Complete integral: w = ϕ(x)(C 1 y + C 2 )–ϕ(x)  F  x, C 1 ϕ(x)  dx ϕ(x) ,whereϕ(x)=exp   g(x) dx  . 8. F  ∂w ∂x , ∂w ∂y  =0. Complete integral: w = C 1 x + C 2 y + C 3 , where C 1 and C 3 are arbitrary constants and the constant C 2 is related toC 1 by F (C 1 , C 2 )=0. 9. w = x ∂w ∂x + y ∂w ∂y + F  ∂w ∂x , ∂w ∂y  . Clairaut’s equation. Complete integral: w = C 1 x + C 2 y + F (C 1 , C 2 ). . C 2 . The signs before each of the integrals can be chosen independently of each other. 6.  ∂w ∂x  2 +  ∂w ∂y  2 = f (x 2 + y 2 ). Hamilton’s equation for the plane motion of a point mass. a  ∂w ∂x  2 + b  ∂w ∂y  2 = c. For a = b,thisisadifferential equation of light rays. Complete integral: w = C 1 x + C 2 y + C 3 ,whereaC 2 1 + bC 2 2 = c. An alternative form of the complete integral: w 2 c = (x. Equations of the Form ∂w ∂x + f(x, y, w) ∂w ∂y = g(x, y, w)  In the solutions of equations T7.2.3.1–T7.2.3.11, Φ(z) is an arbitrary composite function whose argument z can depend on x, y, and w. 1. ∂w ∂x +