dy + P (x)y = Q(x) (1) dx dy dx = P (x)y + Q(x) P (x), Q(x) dy y dx Q(x) 0 homogeneous dy dx = P (x)y 1 y dy = P (x) dx log y = P (x) dx + C y = C exp

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+ P (x)y = Q(x) (1) = P (x)y + Q(x) P (x), Q(x) y Q(x) 0 homogeneous = P (x)y 1 y = P (x) log y = P (x) + C y = C exp{ P (x) } = C e R P (x) 5.1 + P (x)y = 0 (2) y = C exp{ P (x) } = Ce R P (x) (3) αy = 0 (α ) (4) y = Ce αx (5) 18

Q(x) 0 (1) inhomogeneous P (x) α + αy = Q(x) (1.23) 5.2 α αy = Q(x) (6) ( ) y = e αx e αx Q(x) + C = Ce αx + e αx e αx Q(x) (7) 1 y = e αx (e αx y) (6) = d {eαx (e αx y)} = αe αx (e αx y) + e αx d {e αx y} = αy + e αx d {e αx y} e αx d {e αx y} = Q(x) d {e αx y} = e αx Q(x) x e αx y = e αx Q(x) + C 1 y α α α 19

( ) y = e αx e αx Q(x) + C = Ce αx + e αx e αx Q(x) 5.1 2y = x (7) y = Ce 2x + e 2x xe 2x. 2 { xe 2x = x 1 } 2 e 2x ( = {x} 1 ) 2 e 2x + x ( 12 ) e 2x = 1 e 2x x 2 2 e 2x = 1 4 e 2x x 2 e 2x = 1 4 e 2x (1 + 2x). y = Ce 2x + e ( 2x 1 ) 4 e 2x (1 + 2x) = Ce 2x 1 + 2x 4. 2 pp.116-120. 20

(7) C y = e αx e αx Q(x) 1 Q(x) = eαx D α e αx Q(x) (8) (6) α = 0 1 D Q(x) = Q(x) (9) 1 D Ce αx (6) Q(x) (7) = + 5.1 (1) (3) (5) + 2y = 0 (2) 3y = x 2y = ex (4) + y = x2 + 2y = sin x (6) 2y = cos x 5.2 P (x) + P (x)y = Q(x) 21

5.3 + P (x)y = Q(x) (10) { y = e R P (x) = Ce R P (x) + e R P (x) } e R P (x) Q(x) + C e R P (x) Q(x) (11) 3 e R P (x) e R P (x) Q(x) (10) = d R P (x) {e (e R P (x) y)} = P (x)e R P (x) (e R P (x) y) + e R P (x) d {er P (x) y} = P (x)y + e R P (x) d {er P (x) y} (10) e R P (x) d {er P (x) y} = Q(x) d {er P (x) y} = e R P (x) Q(x) x e R P (x) y = e R P (x) Q(x) + C { y = e R P (x) = Ce R P (x) + e R P (x) } e R P (x) Q(x) + C e R P (x) Q(x). 3 Q(x) = 0 (7) 22

4 e R P (x) e R P (x) P (x) 5.2 x + y = 4x(1 + x2 ) x + y x = 4(1 + x2 ) P (x) = 1 x, Q(x) = 4(1 + x2 ) 1 = log x x e log = e R P (x) e R P (x) = e log x = e log x 1 = 1 x, = x. (11) y = C x + 1 4 x (1 + x 2 ) x = C x + 1 4x(1 + x 2 ) x = C x + 1 x (2x2 + x 4 ) = C x + 2x + x3. 1 x x 4 x (1+x2 ) x x = 0 4 p.27 23

5.2 5.3 (11) 5.4 (1) x + (1 + x)y = ex (2) x + y = x log x (3) + 2y tan x = sin x (4) sin x cos x + y = 2 tan x (5) (1 + x 2 ) = 2(1 xy) (6) x2 y + y + 1 + (x + x 3 ) = 0 y 1 (x), y 2 (x) + P (x)y = 0 c 1 y 1 (x) + c 2 y 2 (x) (c 1, c 2 ) d {c 1 1y 1 + c 2 y 2 } = c 1 + c 2 2 = c 1 ( P (x)y 1 ) + c 2 ( P (x)y 2 ) = P (x)(c 1 y 1 + c 2 y 2 ) d {c 1y 1 + c 2 y 2 } + P (x)(c 1 y 1 + c 2 y 2 ) = 0 c 1 y 1 + c 2 y 2 y 1 (x), y 2 (x) + P (x)y = Q(x) ( ) d {y 2 2 y 1 } + P (x)(y 2 y 1 ) = + P (x)y 2 = Q(x) Q(x) = 0 ( ) 1 + P (x)y 1 24

y 2 y 1 y 2 (x) y 1 (x) = C 0 e R P (x) y 2 (x) = y 1 (x) + C 0 e R P (x) 5.3 5.3 + y x = x2 y 3 y z = y 1 3 = y 2 dz = 2y 3. 2y 3 3 2y y 2y 3 x = 2y 3 x 2 y 3, dz 2 x z = 2x2 z 5.3 (11) z = e R 2 x ( 2 = x ( 2 2 = x 2 ( 2x + C) = 2x 3 + Cx 2. ) x 2 e R 2 x + C ) x 2 x 2 + C z = y 2 2x 3 y 2 + cx 2 y 2 = 1. 25

5.5 + P (x)y = Q(x)yα (α 0, 1) (12) z = y 1 α (12) Bernoulli z = y 1 α x dz = (1 α)y α. (1 α)y α (12) (12) α (1 α)y + (1 α)p (x)y1 α = (1 α)q(x). dz + (1 α)p (x)z = (1 α)q(x) z 5.3 (1) 3 + xy = x y 2 (2) x 2 xy + y2 = 0 (3) x 3 = x2 y y 4 cos x (4) = x2 y 6 y x 26

5.6 = f(x) + g(x)y + h(x)y2 (13) 5 (13) Riccati y = ϕ(x) (13) ϕ (x) = f(x) + g(x)ϕ(x) + h(x){ϕ(x)} 2. u = y ϕ(x) x du = ϕ (x) = ( f(x) + g(x)ϕ(x) + h(x){ϕ(x)} 2) = f(x) + g(x)y + h(x)y 2 ( f(x) + g(x)ϕ(x) + h(x){ϕ(x)} 2) = g(x)(y ϕ(x)) + h(x)(y 2 {ϕ(x)} 2 ) = g(x)(y ϕ(x)) + h(x)(y ϕ(x))(y + ϕ(x)) = g(x)u + h(x)u(u + 2ϕ(x)) du (g(x) + 2h(x)ϕ(x))u = h(x)u2. (12) α = 2 z = u 1 5.4 x y + 2y2 = 2x 2 = 2x + y x 2y2 x 5 (13) h(x) 0 f(x) 0 27

f(x) = 2x, g(x) = 1 x, h(x) = 2 x (13) y = x u = y x ( du + 4 1 ) u = 2 x x u2 z = u 1 ( dz 4 1 ) z = 2 x x 28

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