Question

Diberikan nilai awal y(0) = -1/(√2)
, dengan metoda pemisahan variabel selesaikan persamaan berikut :
y' = [ x (x² + 1) ] / 4y³.



@Wello1☺☺...

Answer 1

         y' = [ x (x² + 1) ] / 4y³ ; y(0) = -1/√2
   dy/dx = [ x (x² + 1) ] / 4y³
∫ 4y³ dy = ∫ (x³+x) dx
         y⁴ = (1/4)x⁴ + (1/2)x² + C
          y = ((1/4)x⁴ + (1/2)x² + C)^(1/4)

subtitusi y(0) = -1/√2
      y(x) = ((1/4)x⁴ + (1/2)x² + C)^(1/4)
(-1/√2)⁴ = (1/4)(0)⁴ + (1/2)(0)² + C
       1/4 = C

maka:
y(x) = ((1/4)x⁴ + (1/2)x² + (1/4))^(1/4)
y(x) = ((1/4)(x⁴ + 2x² + 1))^(1/4)
y(x) = -((1/2)(x²+1))^(1/2) atau ((1/2)(x²+1))^(1/2) (ambil yang negatif agar sesuai)
y(x) = -[√(x²+1)]/(√2)