Řešte lineární diferenciální rovnici prvního řádu y + a(x)y = b(x):
1) (x + 1)y - 2y = (x + 1)4 x > -1
[y = (x2
2 + x + c)(x + 1)2
]
2) (2x + 1)y + y =

2x + 1 + 3
[y = 1
2x+1
(c + x + 3 

2x + 1)]
3) y cos x + y sin x = 1 x  (-
2 , 
2 )
[y = (c + tg x)  cos x]
4) y + 1
x+1  y = ln(x + 1) x > -1
[y = x+1
2  ln(x + 1) - x+1
4 + c
x+1 ]
5) y + y cos x = 2 sin x cos x
[y = 2(-1 + sin x) + ce- sin x
]
6) y cos x = (y + 2 cos x) sin x x  (-
2 , 
2 )
[y = 1
2
sin2
x
cos x - cos x + c
cos x ]
7) y = 1 + 2x-1
x2  y
[y = x2
+ cx2
e
1
x ]
8) y + y = x
[y = x - 1 + ce-x
]
9) y + y = cos x
[y = 1
2 (sin x + cos x) + c  e-x
]
10) xy + y = x2 x > 0
[y = x2
3 + c
x ]
11) (1 + x2)  y - xy = 1
[y = x + c 

x2 + 1]
12) xy + 2y = e-x2
[y = 1
2x2 c - e-x2
]
13) y cos x - y sin x = sin 2x x  (-
2 , 
2 )
[y = 1
cos x c - cos 2x
2 ]
14) xy - y
x+1 = x x > 0
[y = x
x+1 (c + x + ln x)]
15) y + x
x2-1
y = arcsin x
1-x2 , x  (-1, 1)
[y = 1
2

1-x2
[c + arcsin2
x]]
16) y - y = 1+x2
x ex
[y = ex
(c + x2
2 + ln |x|)]
17) y - y tg x = sin x
[y = -cos x
2 + c
cos x ]
Robert Mařík,www.mendelu.cz/~marik Typeset by AMS-TEX