Question

Решить дифф. уравнение.
xy'-3y=4x^3

Answer 1

$xy'-3y=4x^3 \ y'-frac{3y}{x}=4x^2 \ y=uv;y'=u'v+uv' \ u'v+uv'-frac{3uv}{x}=4x^2 \ left { {{v'-frac{3v}{x}=0} atop {u'v=4x^2}} right. \ frac{dv}{dx}=frac{3v}{x} \ frac{dv}{v}=3frac{dx}{x} \ lnv=3lnx \ v=x^3 \ frac{du}{dx}x^3=4x^2 \ du=4frac{dx}{x} \ u=4lnx +C\ y=uv=x^3(4lnx+C)$