T7.1. LINEAR EQUATIONS 1249 4. a ∂w ∂x + b ∂w ∂y = f(x) + g(y). General solution: w = 1 a f(x) dx + 1 b g(y) dy + Φ(bx – ay). 5. ∂w ∂x + a ∂w ∂y = f(x)g(y). General solution: w = x x 0 f(t)g(y – ax + at) dt + Φ(y – ax), where x 0 can be taken arbitrarily. 6. ∂w ∂x + a ∂w ∂y = f(x, y). General solution: w = x x 0 f(t, y –ax+at) dt+Φ(y –ax), where x 0 can be taken arbitrarily. 7. ∂w ∂x + [ay + f (x)] ∂w ∂y = g(x). General solution: w = g(x) dx + Φ(u), where u = e –ax y – f(x)e –ax dx. 8. ∂w ∂x + ay + f(x) ∂w ∂y = g(x)h(y). General solution: w = g(x) h e ax u + e ax f(x)e –ax dx dx + Φ(u), where u = e –ax y – f(x)e –ax dx. In the integration, u is treated as a parameter. 9. ∂w ∂x + f(x)y + g(x)y k ∂w ∂y = h(x). General solution: w = h(x) dx + Φ(u), where u = e –F y 1–k –(1 – k) e –F g(x) dx, F =(1 – k) f(x) dx. 10. ∂w ∂x + f(x) + g(x)e λy ∂w ∂y = h(x). General solution: w = h(x) dx + Φ(u), where u = e –λy F (x)+λ g(x)F(x) dx, F (x)=exp λ f(x) dx . 1250 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 11. ax ∂w ∂x + by ∂w ∂y = f(x, y). General solution: w = 1 a 1 x f x, u 1/a x b/a dx + Φ(u), where u = y a x –b . In the integration, u is treated as a parameter. 12. f(x) ∂w ∂x + g(y) ∂w ∂y = h 1 (x) + h 2 (y). General solution: w = h 1 (x) f(x) dx + h 2 (y) g(y) dy + Φ dx f(x) – dy g(y) . 13. f(x) ∂w ∂x + g(y) ∂w ∂y = h(x, y). The transformation ξ = dx f(x) , η = dy g(y) leads to an equation of the form T7.1.2.6 for w = w(ξ, η). 14. f(y) ∂w ∂x + g(x) ∂w ∂y = h(x, y). The transformation ξ = g(x) dx, η = f(y) dy leads to an equation of the form T7.1.2.6 for w = w(ξ, η). T7.1.3. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y)w + r(x, y) In the solutions of equations T7.1.3.1–T7.1.3.10, 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)w. General solution: w =exp 1 a f(x) dx Φ(bx – ay). 2. a ∂w ∂x + b ∂w ∂y = f(x)w + g(x). General solution: w =exp 1 a f(x) dx Φ(bx–ay)+ 1 a g(x)exp – 1 a f(x) dx dx . 3. a ∂w ∂x + b ∂w ∂y = f(x) + g(y) w. General solution: w =exp 1 a f(x) dx + 1 b g(y) dy Φ(bx – ay). T7.1. LINEAR EQUATIONS 1251 4. ∂w ∂x + a ∂w ∂y = f(x, y)w. General solution: w =exp x x 0 f(t, y – ax + at) dt Φ(y – ax), where x 0 can be taken arbitrarily. 5. ∂w ∂x + a ∂w ∂y = f(x, y)w + g(x, y). General solution: w = F (x, u) Φ(u)+ g(x, u + ax) F (x, u) dx , F (x, u)=exp f(x, u + ax) dx , where u = y – ax. In the integration, u is treated as a parameter. 6. ax ∂w ∂x + by ∂w ∂y = f(x)w + g(x). General solution: w =exp 1 a f(x) dx x Φ x –b/a y + 1 a g(x) x exp – 1 a f(x) dx x dx . 7. ax ∂w ∂x + by ∂w ∂y = f(x, y)w. General solution: w =exp 1 a 1 x f x, u 1/a x b/a dx Φ(u), where u = y a x –b . In the integration, u is treated as a parameter. 8. x ∂w ∂x + ay ∂w ∂y = f(x, y)w + g(x, y). General solution: w = F (x, u) Φ(u)+ g(x, ux a ) xF (x, u) dx , F (x, u)=exp 1 x f(x, ux a ) dx , where u = yx –a . In the integration, u is treated as a parameter. 9. f(x) ∂w ∂x + g(y) ∂w ∂y = h 1 (x) + h 2 (y) w. General solution: w =exp h 1 (x) f(x) dx + h 2 (y) g(y) dy Φ dx f(x) dx – dy g(y) dy . 1252 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 10. f 1 (x) ∂w ∂x + f 2 (y) ∂w ∂y = aw + g 1 (x) + g 2 (y). General solution: w = E 1 (x)Φ(u)+E 1 (x) g 1 (x) dx f 1 (x)E 1 (x) + E 2 (y) g 2 (y) dy f 2 (y)E 2 (y) , where E 1 (x)=exp a dx f 1 (x) , E 2 (y)=exp a dy f 2 (y) , u = dx f 1 (x) – dy f 2 (y) . 11. f(x) ∂w ∂x + g(y) ∂w ∂y = h(x, y)w + r(x, y). The transformation ξ = dx f(x) , η = dy g(y) leads to an equation of the form T7.1.3.5 for w = w(ξ, η). 12. f(y) ∂w ∂x + g(x) ∂w ∂y = h(x, y)w + r(x, y). The transformation ξ = g(x) dx, η = f(y) dy leads to an equation of the form T7.1.3.5 for w = w(ξ, η). T7.2. Quasilinear Equations T7.2.1. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y, w) In the solutions of equations T7.2.1.1–T7.2.1.12, Φ(z) is an arbitrary composite function whose argument z can depend on both x and y. 1. ∂w ∂x + a ∂w ∂y = f(x)w + g(x)w k . General solution: w 1–k = F (x)Φ(y – ax)+(1 – k)F (x) g(x) F (x) dx,whereF (x)=exp (1 – k) f(x) dx . 2. ∂w ∂x + a ∂w ∂y = f(x) + g(x)e λw . General solution: e –λw = F (x)Φ(y – ax)–λF (x) g(x) F (x) dx,whereF (x)=exp –λ f(x) dx . T7.2. QUASILINEAR EQUATIONS 1253 3. a ∂w ∂x + b ∂w ∂y = f(w). General solution: dw f(w) = x a + Φ(bx – ay). 4. a ∂w ∂x + b ∂w ∂y = f(x)g(w). General solution: dw g(w) = 1 a f(x) dx + Φ(bx – ay). 5. ∂w ∂x + a ∂w ∂y = f(x)g(y)h(w). General solution: dw h(w) = x x 0 f(t)g(y – ax +at) dt + Φ(y – ax), where x 0 can be taken arbitrarily. 6. ax ∂w ∂x + by ∂w ∂y = f(w). General solution: dw f(w) = 1 a ln |x| + Φ |x| b |y| –a . 7. ay ∂w ∂x + bx ∂w ∂y = f(w). General solution: dw f(w) = 1 √ ab ln √ ab x + ay + Φ ay 2 – bx 2 , ab > 0. 8. ax n ∂w ∂x + by k ∂w ∂y = f(w). General solution: dw f(w) = 1 a(1 – n) x 1–n +Φ(u), where u = 1 a(1 – n) x 1–n – 1 b(1 – k) y 1–k . 9. ay n ∂w ∂x + bx k ∂w ∂y = f(w). General solution: a dw f(w) = b a n + 1 k + 1 x k+1 – u – n n+1 dx,whereu = b a n + 1 k + 1 x k+1 – y n+1 . In the integration, u is treated as a parameter. 10. ae λx ∂w ∂x + be βy ∂w ∂y = f(w). General solution: dw f(w) =– 1 aλ e –λx + Φ(u), where u = aλe –βy – bβe –λx . 1254 FIRST-ORDER PARTIAL DIFFERENTIAL EQUATIONS 11. ae λy ∂w ∂x + be βx ∂w ∂y = f(w). General solution: dw f(w) = c(βx – λy) u + Φ(u), where u = aβe λy – bλe βx . 12. f(x) ∂w ∂x + g(y) ∂w ∂y = h(w). General solution: dw h(w) = dx f(x) + Φ(u), where u = dx f(x) – dy g(y) . 13. f(y) ∂w ∂x + g(x) ∂w ∂y = h(w). The transformation ξ = g(x) dx, η = f(y) dy leads to an equation of the form T7.2.1.5: ∂w ∂ξ + ∂w ∂η = F (ξ)G(η)h(w), where F (ξ)= 1 g(x) , G(η)= 1 f(y) . T7.2.2. Equations of the Form ∂w ∂x + f(x, y, w) ∂w ∂y =0 In the solutions of equations T7.2.2.1–T7.2.2.10, Φ(w) is an arbitrary function. 1. ∂w ∂x + aw + yf(x) ∂w ∂y =0. General solution: yF(x)–aw F (x) dx = Φ(w), where F (x)=exp – f(x) dx . 2. ∂w ∂x + aw + f(y) ∂w ∂y =0. General solution: x = y y 0 dt f(t)+aw + Φ(w). 3. ∂w ∂x + f(w) ∂w ∂y =0. A model equation of gas dynamics. This equation is also encountered in hydrodynamics, multiphase flows, wave theory, acoustics, chemical engineering, and other applications. 1 ◦ . General solution: y = xf(w)+Φ(w), where Φ is an arbitrary function. 2 ◦ . The solution of the Cauchy problem with the initial condition w = ϕ(y)atx = 0 can be represented in the parametric form y = ξ + F(ξ)x, w = ϕ(ξ), where F(ξ)=f ϕ(ξ) . T7.2. QUASILINEAR EQUATIONS 1255 3 ◦ . Consider the Cauchy problem with the discontinuous initial condition w(0, y)= w 1 for y < 0, w 2 for y > 0. It is assumed that x ≥ 0, f > 0,andf > 0 for w > 0, w 1 > 0,andw 2 > 0. Generalized solution for w 1 < w 2 : w(x, y)= w 1 for y/x < V 1 , f –1 (y/x)forV 1 ≤ y/x ≤ V 2 , w 2 for y/x > V 2 , where V 1 = f(w 1 ), V 2 = f (w 2 ). Here f –1 is the inverse of the function f, i.e., f –1 f(w) ≡ w. This solution is continuous in the half-plane x > 0 and describes a “rarefaction wave.” Generalized solution for w 1 > w 2 : w(x, y)= w 1 for y/x < V , w 2 for y/x > V , where V = 1 w 2 – w 1 w 2 w 1 f(w) dw. This solution undergoes a discontinuity along the line y = Vxand describes a “shock wave.” 4 ◦ . In Subsection 13.1.3, qualitative features of solutions to this equation are considered, including the wave-breaking effect and shock waves. This subsection also presents general formulas that permit one to construct generalized (discontinuous) solutions for arbitrary initial conditions. 4. ∂w ∂x + f(w) + ax ∂w ∂y =0. General solution: y = xf (w)+ 1 2 ax 2 + Φ(w). 5. ∂w ∂x + f(w) + ay ∂w ∂y =0. General solution: x = 1 a ln ay + f(w) + Φ(w). 6. ∂w ∂x + f(w) + g(x) ∂w ∂y =0. General solution: y = xf (w)+ g(x) dx + Φ(w). 7. ∂w ∂x + f(w) + g(y) ∂w ∂y =0. General solution: x = y y 0 dt g(t)+f(w) + Φ(w). 8. ∂w ∂x + yf(w) + g(x) ∂w ∂y =0. General solution: y exp –xf(w) – x x 0 g(t)exp –tf(w) dt = Φ(w), where x 0 can be taken arbitrarily. . y)= w 1 for y < 0, w 2 for y > 0. It is assumed that x ≥ 0, f > 0,andf > 0 for w > 0, w 1 > 0,andw 2 > 0. Generalized solution for w 1 < w 2 : w(x, y)= w 1 for y/x. leads to an equation of the form T7.1.3.5 for w = w(ξ, η). T7.2. Quasilinear Equations T7.2.1. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y, w) In the solutions of equations T7.2.1.1–T7.2.1.12,. = f(y) dy leads to an equation of the form T7.1.2.6 for w = w(ξ, η). T7.1.3. Equations of the Form f(x, y) ∂w ∂x + g(x, y) ∂w ∂y = h(x, y)w + r(x, y) In the solutions of equations T7.1.3.1–T7.1.3.10,