PERSAMAAN DIFERENSIAL YANG DAPAT DIPISAH

LATIHAN SOAL

PERSAMAAN DIFERENSIAL YANG DAPAT DIPISAH

1.      (1 + 2x²)y. y’ = 2x (1 + y²)

2.      x2y dx + (x +1) dy

3.      y’ + (y+1) cos x = 0

4.      sin x . cos y dx + tan y . cos x dy = 0

5.    

6.      (x² + 4) = (y + 2) (x + )

7.       =

8.      2xy (4 – y²) dx + ( y – 1) (x² + 2) dy = 0

Jawab:

1.      (1 + 2x²)y. y’ = 2x (1 + y²)

Penyelesaian:

(1 + 2x²)y. y’ = 2x (1 + y²)

(1 + 2x²)y. = 2x (1 + y²)

(1 + 2x²)y. dy = 2x (1 + y²) dx

(1 + 2x²)y. dy - 2x (1 + y²) dx = 0

            Mis:                             mis :

            u = 1 + y2                    u = 1 + 2x²

                                  

            du = 2y dy                  du = 4x dx

                            

                      mis : 2C₁= ln C

 =

 =

 = C

            =  C .

2.      x²y dx + (x +1) dy = 0

Penyelesaian :

x²y dx + (x +1) dy = 0

 

             mis : 2C₁ = ln C

3.      y’ + (y+1) cos x = 0

Penyelesaian:

y’ + (y+1) cos x = 0

         mis: C₁ = ln C

4.      sin x . cos y dx + tan y . cos x dy = 0

Penyelesaian:

sin x . cos y dx + tan y . cos x dy = 0

Mis:                             Mis:                

u = cos x                      u = cos y

                                                                                                                                                                                                                                                                                                                                                                                         

       

                  

          . du

                     du =

                = ln cos x

                     = sec y
                                                            mis: C₁= ln C

5.    

Penyelesaian:

Mis:                            mis:

             

                    

            

            

         mis: 2C₁=ln C

6.      (x² + 4) = (y + 2) (x + )

Penyelesaian:

(x² + 4) = (y + 2) (x + )

Misal  u=                     u=

          du=                   du=

      Misal C₁=ln C

7.     

Penyelesaian:

Untuk:           

untuk: Misal:  u =2

=C

8.      2xy (4 – y²) dx + ( y – 1) (x² + 2) dy = 0

Penyelesaian:

2xy (4 – y²) dx + ( y – 1) (x² + 2) dy = 0

mis:

4A = -1...............................................................(1)

A = -

2B + 2C = 1 ........................................................(2)

-(A + B – C) = 0 ...................................................(3)

Subsitusikan  A = -  ke persamaan (3)

-(A + B – C) = 0            - A – B + C = 0

- B + C = A

- B + C = - .................(4)

Dari persamaan (2) dan (4)

2B + 2C = 1   x 1    2B + 2C = 1

       - B + C = -    x 2   -2B + 2C = -
                                     -
        4B               =

B               =        

Subsitusikan B =  ke persamaan (2)
2B + 2C = 1
2 + 2C = 1
         2C = 1 - 

           C =

Dengan mensubsitusikan nilai A, B dan C. Maka :

Jadi,

    mis: