xy^2*(dy/dx)=y^3-x^3, y(1)=2 (its a initial value problem) Any kind soul here who can give the solution of this eqn?

Damaris Bradshaw

Damaris Bradshaw

Answered question

2023-02-13

x y 2 ( d y d x ) = y 3 - x 3 , y ( 1 ) = 2 (its a initial value problem) Any kind soul here who can give the solution of this eqn?

Answer & Explanation

Shaylee Riggs

Shaylee Riggs

Beginner2023-02-14Added 5 answers

Changing the variable
y = λ x y = x λ + λ we have
x 3 λ 2 ( x λ + λ ) = x 3 λ 3 - x 3 or
λ λ 2 x + 1 = 0
This differential equation can be separated. Combining variables
λ 2 d λ = - d x x and integrating
1 3 λ 3 = - log e x + C 0 or
λ = 1 3 ( C 0 - log e x ) 3 or
y = x 1 3 ( C 0 - log e x ) 3
but y ( 1 ) = 2 = 1 × C 0 3 3 C 0 = 24

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