T6.1. LINEAR SYSTEMS OF TWO EQUATIONS 1235 where C 1 and C 2 are arbitrary constants. On substituting (2) into (1) and integrating, one arrives at the general solution of the original system in the form x = C 3 t + t u(t) t 2 dt, y = C 4 t + t v(t) t 2 dt, where C 3 and C 4 are arbitrary constants. 6. x tt = f (t)(a 1 x + b 1 y), y tt = f (t)(a 2 x + b 2 y). Let k 1 and k 2 be roots of the quadratic equation k 2 –(a 1 + b 2 )k + a 1 b 2 – a 2 b 1 = 0. Then, on multiplying the equations of the system by appropriate constants and on adding them together, one can rewrite the system in the form of two independent equations: z 1 = k 1 f(t)z 1 , z 1 = a 2 x +(k 1 – a 1 )y; z 2 = k 2 f(t)z 2 , z 2 = a 2 x +(k 2 – a 1 )y. Here, a prime stands for a derivative with respect to t. 7. x tt = f (t)(a 1 x t + b 1 y t ), y tt = f(t)(a 2 x t + b 2 y t ). Let k 1 and k 2 be roots of the quadratic equation k 2 –(a 1 + b 2 )k + a 1 b 2 – a 2 b 1 = 0. Then, on multiplying the equations of the system by appropriate constants and on adding them together, one can reduce the system to two independent equations: z 1 = k 1 f(t)z 1 , z 1 = a 2 x +(k 1 – a 1 )y; z 2 = k 2 f(t)z 2 , z 2 = a 2 x +(k 2 – a 1 )y. Integrating these equations and returning to the original variables, one arrives at a linear algebraic system for the unknowns x and y: a 2 x +(k 1 – a 1 )y = C 1 exp k 1 F (t) dt + C 2 , a 2 x +(k 2 – a 1 )y = C 3 exp k 2 F (t) dt + C 4 , where C 1 , , C 4 are arbitrary constants and F(t)= f(t) dt. 8. x tt = af (t)(ty t – y), y tt = bf (t)(tx t – x). The transformation u = tx t – x, v = ty t – y (1) leads to a system of first-order equations: u t = atf(t)v, v t = btf(t)u. 1236 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS The general solution of this system is expressed as if ab > 0, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u(t)=C 1 a exp √ ab tf(t) dt + C 2 a exp – √ ab tf(t) dt , v(t)=C 1 √ ab exp √ ab tf(t) dt – C 2 √ ab exp – √ ab tf(t) dt ; if ab < 0, ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ u(t)=C 1 a cos |ab| tf(t) dt + C 2 a sin |ab| tf(t) dt , v(t)=–C 1 |ab| sin |ab| tf(t) dt + C 2 |ab| cos |ab| tf(t) dt , (2) where C 1 and C 2 are arbitrary constants. On substituting (2) into (1) and integrating, one obtains the general solution of the original system x = C 3 t + t u(t) t 2 dt, y = C 4 t + t v(t) t 2 dt, where C 3 and C 4 are arbitrary constants. 9. t 2 x tt + a 1 tx t + b 1 ty t + c 1 x + d 1 y =0, t 2 y tt + a 2 tx t + b 2 ty t + c 2 x + d 2 y =0. Linear system homogeneous in the independent variable (an Euler type system). 1 ◦ . The general solution is determined by a linear combination of linearly independent particular solutions that are sought by the method of undetermined coefficients in the form of power-law functions x = A|t| k , y = B|t| k . On substituting these expressions into the system and on collecting the coefficients of the unknowns A and B, one obtains [k 2 +(a 1 – 1)k + c 1 ]A +(b 1 k + d 1 )B = 0, (a 2 k + c 2 )A +[k 2 +(b 2 – 1)k + d 2 ]B = 0. For a nontrivial solution to exist, the determinant of this system must vanish. This require- ment results in the characteristic equation [k 2 +(a 1 – 1)k + c 1 ][k 2 +(b 2 – 1)k + d 2 ]–(b 1 k + d 1 )(a 2 k + c 2 )=0, whichisusedtodeterminek. If the roots of this equation, k 1 , , k 4 , are all distinct, then the general solution of the system of differential equations in question has the form x =–C 1 (b 1 k 1 + d 1 )|t| k 1 – C 2 (b 1 k 2 + d 1 )|t| k 2 – C 3 (b 1 k 1 + d 1 )|t| k 3 – C 4 (b 1 k 4 + d 1 )|t| k 4 , y = C 1 [k 2 1 +(a 1 – 1)k 1 + c 1 ]|t| k 1 + C 2 [k 2 2 +(a 1 – 1)k 2 + c 1 ]|t| k 2 + C 3 [k 2 3 +(a 1 – 1)k 3 + c 1 ]|t| k 3 + C 4 [k 2 4 +(a 1 – 1)k 4 + c 1 ]|t| k 4 , where C 1 , , C 4 are arbitrary constants. 2 ◦ . The substitution t = σe τ (σ ≠ 0) leads to a system of constant-coefficient linear differential equations: x ττ +(a 1 – 1)x τ + b 1 y τ + c 1 x + d 1 y = 0, y ττ + a 2 x τ +(b 2 – 1)y τ + c 2 x + d 2 y = 0. T6.2. LINEAR SYSTEMS OF THREE AND MORE EQUATIONS 1237 10. (αt 2 + βt + γ) 2 x tt = ax + by,(αt 2 + βt + γ) 2 y tt = cx + dy. The transformation τ = dt αt 2 + βt + γ , u = x |αt 2 + βt + γ| , v = y |αt 2 + βt + γ| leads to a constant-coefficient linear system of equations of the form T6.1.2.1: u ττ =(a – αγ + 1 4 β 2 )u + bv, v ττ = cu +(d – αγ + 1 4 β 2 )v. 11. x tt = f(t)(tx t – x) + g(t)(ty t – y), y tt = h(t)(tx t – x) + p(t)(ty t – y). The transformation u = tx t – x, v = ty t – y (1) leads to a linear system of first-order equations u t = tf(t)u + tg(t)v, v t = th(t)u + tp(t)v.(2) In order to find the general solution of this system, it suffices to know its any particular solution (see system T6.1.1.7). For solutions of some systems of the form (2), see systems T6.1.1.3–T6.1.1.6. If all functions in (2) are proportional, that is, f(t)=aϕ(t), g(t)=bϕ(t), h(t)=cϕ(t), p(t)=dϕ(t), then the introduction of the new independent variable τ = tϕ(t) dt leads to a constant- coefficient system of the form T6.1.1.1. 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, on 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, where C 3 and C 4 are arbitrary constants. T6.2. Linear Systems of Three and More Equations 1. x t = ax, y t = bx + cy, z t = dx + ky + pz. Solution: x = C 1 e at , y = bC 1 a – c e at + C 2 e ct , z = C 1 a – p d + bk a – c e at + kC 2 c – p e ct + C 3 e pt , where C 1 , C 2 ,andC 3 are arbitrary constants. 1238 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 2. x t = cy – bz, y t = az – cx, z t = bx – ay. 1 ◦ . First integrals: ax + by + cz = A,(1) x 2 + y 2 + z 2 = B 2 ,(2) where A and B are arbitrary constants. It follows that the integral curves are circles formed by the intersection of planes (1) and spheres (2). 2 ◦ . Solution: x = aC 0 + kC 1 cos(kt)+(cC 2 – bC 3 )sin(kt), y = bC 0 + kC 2 cos(kt)+(aC 3 – cC 1 )sin(kt), z = cC 0 + kC 3 cos(kt)+(bC 1 – aC 2 )sin(kt), where k = √ a 2 + b 2 + c 2 and the three of four constants of integration C 0 , , C 3 are related by the constraint aC 1 + bC 2 + cC 3 = 0. 3. ax t = bc(y – z), by t = ac(z – x), cz t = ab(x – y). 1 ◦ . First integral: a 2 x + b 2 y + c 2 z = A, where A is an arbitrary constant. It follows that the integral curves are plane ones. 2 ◦ . Solution: x = C 0 + kC 1 cos(kt)+a –1 bc(C 2 – C 3 )sin(kt), y = C 0 + kC 2 cos(kt)+ab –1 c(C 3 – C 1 )sin(kt), z = C 0 + kC 3 cos(kt)+abc –1 (C 1 – C 2 )sin(kt), where k = √ a 2 + b 2 + c 2 and the three of four constants of integration C 0 , , C 3 are related by the constraint a 2 C 1 + b 2 C 2 + c 2 C 3 = 0. 4. x t = (a 1 f + g)x + a 2 fy + a 3 fz, y t = b 1 fx + (b 2 f + g)y + b 3 fz, z t = c 1 fx + c 2 fy + (c 3 f + g)z. Here, f = f (t)andg = g(t). The transformation x =exp g(t) dt u, y =exp g(t) dt v, z =exp g(t) dt w, τ = f(t) dt leads to the system of constant coefficient linear differential equations u τ = a 1 u + a 2 v + a 3 w, v τ = b 1 u + b 2 v + b 3 w, w τ = c 1 u + c 2 v + c 3 w. 5. x t = h(t)y – g(t)z, y t = f(t)z – h(t)x, z t = g(t)x – f(t)y. 1 ◦ . First integral: x 2 + y 2 + z 2 = C 2 , where C is an arbitrary constant. 2 ◦ . The system concerned can be reduced to a Riccati equation (see Kamke, 1977). T6.3. NONLINEAR SYSTEMS OF TWO EQUATIONS 1239 6. x k = a k1 x 1 + a k2 x 2 + ···+ a kn x n ; k =1, 2, , n. System of n constant-coefficient first-order linear homogeneous differential equations. The general solution of a homogeneous system of differential equations is determined by a linear combination of linearly independent particular solutions, which are sought by the method of undetermined coefficients in the form of exponential functions, x k = A k e λt ; k = 1, 2, , n. On substituting these expressions into the system and on collecting the coefficients of the unknowns A k , one obtains a linear homogeneous system of algebraic equations: a k1 A 1 + a k2 A 2 + ···+(a kk – λ)A k + ···+ a kn A n = 0; k = 1, 2, , n. For a nontrivial solution to exist, the determinant of this system must vanish. This require- ment results in a characteristic equation that serves to determine λ. T6.3. Nonlinear Systems of Two Equations T6.3.1. Systems of First-Order Equations 1. x t = x n F (x, y), y t = g(y)F (x, y). Solution: x = ϕ(y), dy g(y)F (ϕ(y), y) = t + C 2 , where ϕ(y)= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ C 1 +(1 – n) dy g(y) 1 1–n if n ≠ 1, C 1 exp dy g(y) if n = 1, C 1 and C 2 are arbitrary constants. 2. x t = e λx F (x, y), y t = g(y)F (x, y). Solution: x = ϕ(y), dy g(y)F (ϕ(y), y) = t + C 2 , where ϕ(y)= ⎧ ⎪ ⎨ ⎪ ⎩ – 1 λ ln C 1 – λ dy g(y) if λ ≠ 0, C 1 + dy g(y) if λ = 0, C 1 and C 2 are arbitrary constants. 1240 SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS 3. x t = F (x, y), y t = G(x, y). Autonomous system of general form. Suppose y = y(x, C 1 ), where C 1 is an arbitrary constant, is the general solution of the first-order equation F (x, y)y x = G(x, y). Then the general solution of the system in question results in the following dependence for the variable x: dx F (x, y(x, C 1 )) = t + C 2 . 4. x t = f 1 (x)g 1 (y)Φ(x, y, t), y t = f 2 (x)g 2 (y)Φ(x, y, t). First integral: f 2 (x) f 1 (x) dx – g 1 (y) g 2 (y) dy = C,(∗) where C is an arbitrary constant. On solving (∗)forx (or y) and on substituting the resulting expression into one of the equations of the system concerned, one arrives at a first-order equation for y (or x). 5. x = tx t + F (x t , y t ), y = ty t + G(x t , y t ). Clairaut system. The following are solutions of the system: (i) straight lines x = C 1 t + F(C 1 , C 2 ), y = C 2 t + G(C 1 , C 2 ), where C 1 and C 2 are arbitrary constants; (ii) envelopes of these lines; (iii) continuously differentiable curves that are formed by segments of curves (i) and (ii). T6.3.2. Systems of Second-Order Equations 1. x tt = xf (ax – by) + g(ax – by), y tt = yf(ax – by) + h(ax – by). Let us multiply the fi rst equation by a and the second one by –b and add them together to obtain the autonomous equation z tt = zf(z)+ag(z)–bh(z), z = ax – by.(1) We will consider this equation in conjunction with the first equation of the system, x tt = xf(z)+g(z). (2) Equation (1) can be treated separately; its general solution can be written out in implicit form (see Polyanin and Zaitsev, 2003). The function x = x(t) can be determined by solving the linear equation (2), and the function y = y(t) is found as y =(ax – z)/b. T6.3. NONLINEAR SYSTEMS OF TWO EQUATIONS 1241 2. x tt = xf (y/x), y tt = yg(y/x). A periodic particular solution: x = C 1 sin(kt)+C 2 cos(kt), k = –f(λ), y = λ[C 1 sin(kt)+C 2 cos(kt)], where C 1 and C 2 are arbitrary constants and λ is a root of the transcendental (algebraic) equation f(λ)=g(λ). (1) 2 ◦ . Particular solution: x = C 1 exp(kt)+C 2 exp(–kt), k = f(λ), y = λ[C 1 exp(kt)+C 2 exp(–kt)], where C 1 and C 2 are arbitrary constants and λ is a root of the transcendental (algebraic) equation (1). 3. x tt = kxr –3 , y tt = kyr –3 , where r = x 2 + y 2 . Equation of motion of a point mass in the xy-plane under gravity. Passing to polar coordinates by the formulas x = r cos ϕ, y = r sin ϕ, r = r(t), ϕ = ϕ(t), one may obtain the first integrals r 2 ϕ t = C 1 ,(r t ) 2 + r 2 (ϕ t ) 2 =–2kr –1 + C 2 ,(1) where C 1 and C 2 are arbitrary constants. Assuming that C 1 ≠ 0 and integrating further, one finds that r[C cos(ϕ – ϕ 0 )–k]=C 2 1 , C 2 = C 2 1 C 2 + k 2 . This is an equation of a conic section. The dependence ϕ(t) may be found from the fi rst equation in (1). 4. x tt = xf (r), y tt = yf(r), where r = x 2 + y 2 . Equation of motion of a point mass in the xy-plane under a central force. Passing to polar coordinates by the formulas x = r cos ϕ, y = r sin ϕ, r = r(t), ϕ = ϕ(t), one may obtain the first integrals r 2 ϕ t = C 1 ,(r t ) 2 + r 2 (ϕ t ) 2 = 2 rf(r) dr + C 2 , where C 1 and C 2 are arbitrary constants. Integrating further, one finds that t + C 3 = rdr 2r 2 F (r)+r 2 C 2 – C 2 1 , ϕ = C 1 dt r + C 4 ,(∗) where C 3 and C 4 are arbitrary constants and F (r)= rf(r) dr. It is assumed in the second relation in (∗) that the dependence r = r(t) is obtained by solving the first equation in (∗)forr(t). . SYSTEMS OF TWO EQUATIONS 1235 where C 1 and C 2 are arbitrary constants. On substituting (2) into (1) and integrating, one arrives at the general solution of the original system in the form x. is determined by a linear combination of linearly independent particular solutions that are sought by the method of undetermined coefficients in the form of power-law functions x = A|t| k , y. general solution of this system, it suffices to know its any particular solution (see system T6.1.1.7). For solutions of some systems of the form (2), see systems T6.1.1.3–T6.1.1.6. If all functions