The solution of the equation x^2 y -x^3 dy d x = y ^4 x , where y (0)=1 , is

Step-by-step solution

The differential equation x^2 y - x^3 dy dx = y^4 x can be rearranged to reveal its Bernoulli form. Dividing by -x^3 yields dy dx - 1 x y = - y^4 x^3 x , which corresponds to the Bernoulli equation dy dx + P(x)y = Q(x)y^n with P(x) = - 1 x , Q(x) = - x x^3 , and n = 4 . Applying the substitution v = y^ 1-n = y⁻³ , we differentiate to obtain dv dx = -3y⁻⁴ dy dx , which implies y⁻⁴ dy dx = - 1 3 dv dx . Dividing the original equation by y^4 gives y⁻⁴ dy dx - 1 x y⁻³ = - x x^3 . Substituting v and its derivative yield