persamaan diferensial linier

Selesaikan PD linier berikut !

11.  y’ + 2y = 3e ^-2x 

Jwb :

Y’ + 2y = 3e ^{– 2x}

P(x) = 2 , r(x) = 3e ^{– 2x}

H = integral p(x) dx

    = integral 2 dx

    = 2x

Y = e^-h (integral r dx + c )

    = e^-2x (integral e. 3e^-2x dx + c)

    = e^-2x (integral 3 dx + c)

    = e^-2x ( 3x + c)

    = 3x e^-2x + ce^-2x

 

2. xy’ + y = x³

Y’ + y/x = x²

P(x) =1/x , r(x) = x²

H = integral 1/x

   = ln x

Y= e ^{– ln x} ( integral e ^{ln x} . x² dx )

   = e ^–1 ( integral x . x² dx + c )

   = e ^-1 ( integral x³ dx + c)

   = e ^-1 (1/4 x⁴ + c)

   = ¼ e³ + ce^-1 

3. dy/ dx + 5y = 0

P(x) = 5 , r(x) = 0

H = integral 5 dx = 5x

Y= e ^-5x ( integral e ^5x . 0 dx + c)

   = e ^-5x (integral 0 dx + c)

   = e ^-5x ( c)

   = ce^-5x