 .  .  .
 


              .      ,         .       .       ,    ,     .





 

 .  .  .



:

1)         .  .

2)           .

3)        .  .

4)          .



1))        .

        :

1) .

2) .

1)    :

       ()  ,    (y ')  ,     (y '')  ;      .

         ,  .

           , :

1.   .

.

) y '' - 5y ' + 6y = 0

) y '' -  4y ' = 0

) y '' - 25 y = 0

) y '' + 2y ' + y = 0

) 3 y '' = 0

) y '' + 2y ' + 5y = 0

) y '' + 9 y = 0

   ,       k ?,      k,      .

y '' = k ?

y ' = k

y = 1

) k ? - 5 k + 6 = 0

) k ? - 4 k = 0

) k ? - 25  = 0

) k ? + 2 k + 1 = 0

) 3 k ? = 0

) k ? + 2 k + 5 = 0

) k ? + 9  = 0

2.    .

  -   .

     .

  :

1)     .

2)    .

3)    .

              .

1)        ,  D > 0.

  ,     ,    :

m k ? + p k + q = 0

m ? 0     p ? 0     q ? 0

D = p ? - 4 m q > 0

k1 = (- p + ?D) / 2m

k2 = (- p - ?D) / 2m

   :

1) m k ?  p k = 0

m ? 0     p ? 0     q = 0

k (m k  p)= 0

k = 0

m k  p= 0

m k =  p

k =  p / m

2) m k ? - q = 0

m ? 0     p = 0     q < 0

m k ? = q

k ? = q / m

k1 = ?(q / m)

k2 = - ?(q / m)

) k ? - 5 k + 6 = 0

D = 25 - 24 = 1

k1 = (5 + 1) / 2 = 3

k2 = (5 - 1) / 2 = 2

) k ? - 4 k = 0

k (k - 4) = 0

k = 0

k = 4

) k ? - 25  = 0

k ? = 25

k1 = 5

k2 = - 5

2)           ,      :

m k ? + p k + q = 0

m ? 0     p ? 0     q ? 0

D = p ? - 4 m q = 0

k1 = (- p + ?0) / 2m = -  / 2m

    :

m k ? = 0

m ? 0     p = 0     q = 0

k ? = 0

k = 0

) k ? + 2 k + 1 = 0

D = 4 - 4 = 0

k = - 2 / 2 = - 1

) 3 k ? = 0

k ? = 0

k  = 0

3)        ,      :

m k ? + p k + q = 0

m ? 0     p ? 0     q ? 0

D = p ? - 4 m q < 0

k1 = (- p + ?(- D)) / 2m = ? + ? i

k2 = (- p - ?(- D)) / 2m = ? - ? i








    :

m k ? + q = 0

m ? 0     p = 0     q > 0

m k ? = - q

k ? = - q /m

k1 = ?(- q /m) = 0 + ? i = ? i

k2 = - ?(- q /m) = 0 - ? i = - ? i

) k ? + 2 k + 5 = 0

D = 4 - 20 = - 16

k1 = (- 2 + ?(- 16)) /2 = (- 2 + 4 i) / 2 = - 1 + 2 i

k2 = (- 2 - ?(- 16)) / 2 = (- 2 - 4 i) / 2 = - 1 - 2 i

) k ? + 9  = 0

k ? = - 9

k 1 = ?(- 9)

k 2 = - ?(- 9)

k 1 =  3 i

k 2 = - 3 i

3.       .

    ,          ,   :

m  '' + p  ' + q  = 0

m k ? + p k + q = 0

m ? 0     p ? 0     q ? 0

D = p ? - 4 m q > 0

k1 = (- p + ?D) / 2m

k2 = (- p - ?D) / 2m








 k1; k2    .

   ,    ,   :

m  '' + p  ' = 0

m k ?  p k = 0

m ? 0     p ? 0     q = 0

k (m k  p)= 0

k1 = 0

m k  p= 0

m k =  p

k2 =  p / m

     .








  k2     .

m  '' + q  = 0

m k ? - q = 0

m ? 0     p = 0

q < 0     m > 0;

q > 0     m < 0

m k ? = q

k ? = q  / m

k1 = ?(q  / m)

k2 = - ?(q  / m)








 k1; k2     .

) y '' - 5 y ' + 6 = 0

k ? - 5 k + 6 = 0

D = 25 - 24 = 1

k 1 = (5 + 1) / 2 = 3

k2 = (5 - 1) / 2 = 2








) y '' - 4 y ' = 0

k ? - 4 k = 0

k (k - 4) = 0

k = 0

k = 4








) y ''- 25 = 0

k ? - 25  = 0

k ? = 25

k1 = 5

k2 = - 5








         ,      :

  .

m  '' + p  ' + q  = 0

m k? + p k + q = 0

m ? 0     p ? 0     q ? 0

D = p ? - 4 m q = 0

k1 = (- p + ?0) / 2m = -  / 2m








  .

m  '' = 0

m k ? = 0

m ? 0     p = 0     q = 0

k ? = 0

k = 0








) y '' + 2 y ' + 1 = 0

k ? + 2 k + 1 = 0

D = 4 - 4 = 0

k = - 2 / 2 = - 1








) 3 y '' = 0

3 k ? = 0

k ? = 0

k = 0








    ,    ,   :

  .

m  '' + p  ' + q  = 0

m k ? + p k + q = 0

m ? 0     p ? 0     q ? 0

D = p ? - 4 m q < 0

k1 = (- p + ?(- D)) / 2m = ? + ? i

k2 = (- p - ?(- D)) / 2m = ? - ? i








  .

m  '' + q  = 0

m k ? + q = 0

m ? 0     p = 0

q > 0    m > 0;

q < 0    m < 0

m k ? = - q

k ? = - q /m

k 1 = ?(- q /m) = 0 + ? i = ? i

k 2 = - ?(- q /m) = 0 - ? i = ? i








) y '' + 2 y ' + 5 = 0

k ? + 2 k + 5 = 0

D = 4 - 20 = - 16

k1 = (- 2 + ?(- 16)) /2 = (- 2 + 4) / 2 = - 1 + 2 i

k2 = (- 2 - ?(- 16)) / 2 = (- 2 - 4) / 2 = - 1 - 2 i








) y '' + 9  = 0

k ? + 9  = 0

k ? = - 9

k 1 = ?(- 9)

k 2 = - ?(- 9)

k 1 =  3 i

k 2 = - 3 i

y = C1 sin 3x + C2 cos 3x

4)  .

     ,    ,    .

     ,    ,    .

   ,             .








2))         ( -  x).

 .

y '' + y ' - 2y = 0

  .

k ? + k - 2 = 0

D = 1 + 4 ? 2 = 9 = 3 ?

k1 = (- 1 + 3) / 2 = 1

k2 = (- 1 - 3) / 2 = - 2

: Y = C1 e * + C2e ??*

 .

y '' - 4 y = 0

  .

k ? - 4 = 0

k ? = 4

k1 = 2

k2 = - 2

: y = C1 e ?* + C2 e ??*

 .

y '' - 9 y ' = 0

  .

k ? - 9 k = 0

k (k - 9) = 0

k1 = 0

k - 9 = 0

k2 = 9

:  y = C1 + C2 e ?*

 .

y '' - 4 y ' + 4 = 0

  .

k ? - 4 k + 4 = 0

D = 16 - 16 = 0

k = 4 / 2 = 2

:   y = C1 e ?* + C2 x e ?*

 .

8y '' = 0

  .

8 k ? = 0

k ? = 0

k = 0

:  y = C1 + C2 x

 .

y '' + y = 0

  .

k ? + 1 = 0

k ? = - 1

k1 = ?(- 1)

k1 = i

k2 = - ?(- 1)

k2 = - i

:   y = C1 sin x + C2 cos x

 .

y '' + 6y ' + 13y = 0

  .

k ? + 6 k + 13 = 0

D = 36 - 52 = - 16

k1 = (- 6 + ?(- 16)) / 2 = - 3 + 2 i

k2 = (- 6 - ?(- 16)) / 2 = - 3 - 2 i

:  y = C1 e ??* sin 2x + C2 e ??* cos 2x

 .

4y '' - 8y ' + 5y = 0

  .

4 k ? - 8 k + 5 = 0

D = 64 - 80 = - 16

k1 = (8 + ?(- 16)) / 8 = (8 + 4 i) / 8 = 1 + 1/2 i

k2 = (8 - ?(- 16)) / 8 = (8 - 4 i) / 8 = 1 - 1/2 i

:  y = C1 e * sin 1/2 x + C2 e * cos 1/2 x

 .

y ''' - 5y '' + 4y ' = 0

  .

k ? - 5 k ? + 4 k = 0

  .

 k  .

k (k ? - 5 k + 4) = 0

k = 0

k ? - 5 k + 4 = 0

D = 25 - 16 = 9 = 3 ?

k1 = (5 + 3) / 2 = 4

k2 = (5 - 3) / 2 = 1

:  y = C1 + C2 e ?* + C3  *

   ,     .








 .

y ''' + 9 y ' = 0

  .

k ? + 9 k = 0

  .

 k  .

k (k ? + 9) = 0

k = 0

k ? + 9 = 0

k ? = - 9

k1 = ?(- 9)

k1 = 3 i

k 2 = - ?(- 9)

k2 = - 3 i

    .

:  y = C1 + C2 sin 3x + C3 cos 3x

        . 








 .

y ''' - 13y ' + 12y = 0

  .




  .


   .

   ,     (https://www.litres.ru/pages/biblio_book/?art=74266267)  .

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