1242 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 5. x tt + a(t)x = x –3 f(y/x), y tt + a(t)y = y –3 g(y/x). Generalized Ermakov system. 1 ◦ . First integral: 1 2 (xy t – yx t ) 2 + y/x uf(u)–u –3 g(u) du = C, where C is an arbitrary constant. 2 ◦ . Suppose ϕ = ϕ(t) is a nontrivial solution of the second-order linear differential equation ϕ tt + a(t)ϕ = 0.(1) Then the transformation τ = dt ϕ 2 (t) , u = x ϕ(t) , v = y ϕ(t) (2) leads to the autonomous system of equations u ττ = u –3 f(v/u), v ττ = v –3 g(v/u). (3) 3 ◦ . Particular solution of system (3) is u = A C 2 τ 2 + C 1 τ + C 0 , v = Ak C 2 τ 2 + C 1 τ + C 0 , A = f(k) C 0 C 2 – 1 4 C 2 1 1/4 , where C 0 , C 1 ,andC 2 are arbitrary constants, and k is a root of the algebraic (transcendental) equation k 4 f(k)=g(k). 6. x tt = f(y t /x t ), y tt = g(y t /x t ). 1 ◦ . The transformation u = x t , w = y t (1) leads to a system of the first-order equations u t = f (w/u), w t = g(w/u). (2) Eliminating t yields a homogeneous first-order equation, whose solution is given by f(ξ) dξ g(ξ)–ξf(ξ) =ln|u| + C, ξ = w u ,(3) where C is an arbitrary constant. On solving (3) for w, one obtains w = w(u, C). On substituting this expression into the first equation of (2), one can find u = u(t)andthen w = w(t). Finally, one can determine x = x(t)andy = y(t) from (1) by simple integration. T6.3. NONLINEAR SYSTEMS OF TWO EQUATIONS 1243 2 ◦ . The Suslov problem. The problem of a point particle sliding down an inclined rough plane is described by the equations x tt = 1 – kx t (x t ) 2 +(y t ) 2 , y tt =– ky t (x t ) 2 +(y t ) 2 , which correspond to a special case of the system in question with f(z)=1 – k √ 1 + z 2 , g(z)=– kz √ 1 + z 2 . The solution of the corresponding Cauchy problem under the initial conditions x(0)=y(0)=x t (0)=0, y t (0)=1 leads, for the case k = 1, to the following dependences x(t)andy(t) written in parametric form: x =– 1 16 + 1 16 ξ 4 – 1 4 ln ξ, y = 2 3 – 1 2 ξ – 1 6 ξ 3 , t = 1 4 – 1 4 ξ 2 – 1 2 ln ξ (0 ≤ ξ ≤ 1). 7. x tt = xΦ(x, y, t, x t , y t ), y tt = yΦ(x, y, t, x t , y t ). 1 ◦ . First integral: xy t – yx t = C, where C is an arbitrary constant. Remark. The function Φ can also be dependent on the second and higher derivatives with respect to t. 2 ◦ . Particular solution: y = C 1 x,whereC 1 is an arbitrary constant and the function x = x(t) is determined by the ordinary differential equation x tt = xΦ(x, C 1 x, t, x t , C 1 x t ). 8. x tt + x –3 f(y/x) = xΦ(x, y, t, x t , y t ), y tt + y –3 g(y/x) = yΦ(x, y, t, x t , y t ). First integral: 1 2 (xy t – yx t ) 2 + y/x u –3 g(u)–uf(u) du = C, where C is an arbitrary constant. Remark. The function Φ can also be dependent on the second and higher derivatives with respect to t. 9. x tt = F (t, tx t – x, ty t – y), y tt = G(t, tx t – x, ty t – y). 1 ◦ . The transformation u = tx t – x, v = ty t – y (1) leads to a system of first-order equations u t = tF(t, u, v), v t = tG(t, u, v). (2) 2 ◦ . Suppose a solution of system (2) has been found in the form u = u(t, C 1 , C 2 ), v = v(t, C 1 , C 2 ), (3) where C 1 and C 2 are arbitrary constants. Then, substituting (3) into (1) and integrating, one obtains a solution of the original system, x = C 3 t + t u(t, C 1 , C 2 ) t 2 dt, y = C 4 t + t v(t, C 1 , C 2 ) t 2 dt. 3 ◦ . If the functions F and G are independent of t, then, on eliminating t from system (2), one arrives at a first-order equation g(u, v)u v = F (u, v). 1244 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS T6.4. Nonlinear Systems of Three or More Equations 1. ax t = (b – c)yz, by t = (c – a)zx, cz t = (a – b)xy. First integrals: ax 2 + by 2 + cz 2 = C 1 , a 2 x 2 + b 2 y 2 + c 2 z 2 = C 2 , where C 1 and C 2 are arbitrary constants. On solving the first integrals for y and z and on substituting the resulting expressions into the first equation of the system, one arrives at a separable first-order equation. 2. ax t = (b – c)yzF (x, y, z, t), by t = (c – a)zxF (x, y, z, t), cz t = (a – b)xyF (x, y, z, t). First integrals: ax 2 + by 2 + cz 2 = C 1 , a 2 x 2 + b 2 y 2 + c 2 z 2 = C 2 , where C 1 and C 2 are arbitrary constants. On solving the first integrals for y and z and on substituting the resulting expressions into the first equation of the system, one arrives at a separable first-order equation; if F is independent of t, this equation will be separable. 3. x t = cF 2 – bF 3 , y t = aF 3 – cF 1 , z t = bF 1 – aF 2 , where F n = F n (x, y, z). First integral: ax + by + cz = C 1 , where C 1 is an arbitrary constant. On eliminating t and z from the first two equations of the system (using the above fi rst integral), one arrives at the first-order equation dy dx = aF 3 (x, y, z)–cF 1 (x, y, z) cF 2 (x, y, z)–bF 3 (x, y, z) ,wherez = 1 c (C 1 – ax – by). 4. x t = czF 2 – byF 3 , y t = axF 3 – czF 1 , z t = byF 1 – axF 2 . Here, F n = F n (x, y, z) are arbitrary functions (n = 1, 2, 3). First integral: ax 2 + by 2 + cz 2 = C 1 , where C 1 is an arbitrary constant. On eliminating t and z from the first two equations of the system (using the above fi rst integral), one arrives at the first-order equation dy dx = axF 3 (x, y, z)–czF 1 (x, y, z) czF 2 (x, y, z)–byF 3 (x, y, z) ,wherez = 1 c (C 1 – ax 2 – by 2 ). 5. x t = x(cF 2 – bF 3 ), y t = y(aF 3 – cF 1 ), z t = z(bF 1 – aF 2 ). Here, F n = F n (x, y, z) are arbitrary functions (n = 1, 2, 3). First integral: |x| a |y| b |z| c = C 1 , where C 1 is an arbitrary constant. On eliminating t and z from the first two equations of the system (using the above fi rst integral), one may obtain a first-order equation. T6.4. NONLINEAR SYSTEMS OF THREE OR MORE EQUATIONS 1245 6. x t = h(z)F 2 – g(y)F 3 , y t = f (x)F 3 – h(z)F 1 , z t = g(y)F 1 – f(x)F 2 . Here, F n = F n (x, y, z) are arbitrary functions (n = 1, 2, 3). First integral: f(x) dx + g(y) dy + h(z) dz = C 1 , where C 1 is an arbitrary constant. On eliminating t and z from the first two equations of the system (using the above fi rst integral), one may obtain a first-order equation. 7. x tt = ∂F ∂x , y tt = ∂F ∂y , z tt = ∂F ∂z , where F = F (r), r = x 2 + y 2 + z 2 . Equations of motion of a point particle under gravity. The system can be rewritten as a single vector equation: r tt =gradF or r tt = F (r) r r, where r =(x, y, z). 1 ◦ . First integrals: (r t ) 2 = 2F (r)+C 1 (law of conservation of energy), [r × r t ]=C (law of conservation of areas), (r ⋅ C)=0 (all trajectories are plane curves). 2 ◦ . Solution: r = a r cos ϕ + b r sin ϕ. Here, the constant vectors a and b must satisfy the conditions |a| = |b| = 1,(a ⋅ b)=0, and the functions r = r(t)andϕ = ϕ(t)aregivenby t = rdr 2r 2 F (r)+C 1 r 2 – C 2 3 + C 2 , ϕ = C 3 dr r 2r 2 F (r)+C 1 r 2 – C 2 3 , C 3 = |C|. 8. x tt = xF , y tt = yF , z tt = zF , where F = F (x, y, z, t, x t , y t , z t ). First integrals (laws of conservation of areas): zy t – yz t = C 1 , xz t – zx t = C 2 , yx t – xy t = C 3 , where C 1 , C 2 ,andC 3 are arbitrary constants. Corollary of the conservation laws: C 1 x + C 2 y + C 3 z = 0. This implies that all integral curves are plane ones. Remark. The function Φ can also be dependent on the second and higher derivatives with respect to t. 1246 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 9. x tt = F 1 , y tt = F 2 , z tt = F 3 , where F n = F n (t, tx t – x, ty t – y, tz t – z). 1 ◦ . The transformation u = tx t – x, v = ty t – y, w = tz t – z (1) leads to the system of first-order equations u t = tF 1 (t, u, v, w), v t = tF 2 (t, u, v, w), w t = tF 3 (t, u, v, w). (2) 2 ◦ . Suppose a solution of system (2) has been found in the form u(t)=u(t, C 1 , C 2 , C 3 ), v(t)=v(t, C 1 , C 2 , C 3 ), w(t)=w(t, C 1 , C 2 , C 3 ), (3) where C 1 , C 2 ,andC 3 are arbitrary constants. Then, substituting (3) into (1) and integrating, one obtains a solution of the original system: x = C 4 t + t u(t) t 2 dt, y = C 5 t + t v(t) t 2 dt, z = C 6 t + t w(t) t 2 dt, where C 4 , C 5 ,andC 6 are arbitrary constants. References for Chapter T6 Kamke, E., Differentialgleichungen: L ¨ osungsmethoden undL ¨ osungen, I, Gew ¨ ohnlicheDifferentialgleichungen, B. G. Teubner, Leipzig, 1977. Polyanin, A. D., Systems of Ordinary Differential Equations, From Website EqWorld—The World of Mathe- matical Equations, http://eqworld.ipmnet.ru/en/solutions/sysode.htm. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd Edition, Chapman & Hall/CRC Press, Boca Raton, 2003. Chapter T7 First-Order Partial Differential Equations T7.1. Linear Equations T7.1.1. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y =0 In equations T7.1.1.1–T7.1.1.11, the general solution is expressed in terms of the prin- cipal integral Ξ as w = Φ(Ξ),whereΦ(Ξ) is an arbitrary function. 1. ∂w ∂x + f(x)y + g(x) ∂w ∂y =0. Principal integral: Ξ = e –F y – e –F g(x) dx,whereF = f(x) dx. 2. ∂w ∂x + f(x)y + g(x)y k ∂w ∂y =0. Principal integral: Ξ = e –F y 1–k –(1 – k) e –F g(x) dx,whereF =(1 – k) f(x) dx. 3. ∂w ∂x + f(x)e λy + g(x) ∂w ∂y =0. Principal integral: Ξ = e –λy E + λ f(x)Edx,whereE =exp λ g(x) dx . 4. f(x) ∂w ∂x + g(y) ∂w ∂y =0. Principal integral: Ξ = dx f(x) – dy g(y) . 5. f(y) + amx n y m–1 ∂w ∂x – g(x) + anx n–1 y m ∂w ∂y =0. Principal integral: Ξ = f(y) dy + g(x) dx + ax n y m . 6. e αx f(y) + cβ ∂w ∂x – e βy g(x) + cα ∂w ∂y =0. Principal integral: Ξ = e –βy f(y) dy + e –αx g(x) dx – ce –αx–βy . 1247 1248 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 7. ∂w ∂x + f(ax + by + c) ∂w ∂y =0, b ≠ 0. Principal integral: Ξ = dv a + bf(v) – x,wherev = ax + by + c. 8. ∂w ∂x + f y x ∂w ∂y =0. Principal integral: Ξ = dv f(v)–v –ln|x|,wherev = y x . 9. x ∂w ∂x + yf(x n y m ) ∂w ∂y =0. Principal integral: Ξ = dv v mf(v)+n –ln|x|,wherev = x n y m . 10. ∂w ∂x + yf(e αx y m ) ∂w ∂y =0. Principal integral: Ξ = dv v α + mf (v) – x,wherev = e αx y m . 11. x ∂w ∂x + f(x n e αy ) ∂w ∂y =0. Principal integral: Ξ = dv v n + αf(v) –ln|x|,wherev = x n e αy . T7.1.2. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y) In the solutions of equations T7.1.2.1–T7.1.2.12, Φ(z) is an arbitrary composite function whose argument z can depend on both x and y. 1. a ∂w ∂x + b ∂w ∂y = f (x). General solution: w = 1 a f(x) dx + Φ(bx – ay). 2. ∂w ∂x + a ∂w ∂y = f (x)y k . General solution: w = x x 0 (y–ax+at) k f(t) dt+Φ(y–ax), where x 0 can be taken arbitrarily. 3. ∂w ∂x + a ∂w ∂y = f (x)e λy . General solution: w = e λ(y–ax) f(x)e aλx dx + Φ(y – ax). . Website EqWorld—The World of Mathe- matical Equations, http://eqworld.ipmnet.ru/en/solutions/sysode.htm. Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential. c 2 z 2 = C 2 , where C 1 and C 2 are arbitrary constants. On solving the first integrals for y and z and on substituting the resulting expressions into the first equation of the system, one arrives. C 2 are arbitrary constants. On solving the first integrals for y and z and on substituting the resulting expressions into the first equation of the system, one arrives at a separable first-order equation. 2.