Question

решите дифференциальное уравнение xy''-9y'=x

Answer 1

$xy''-9y'=x\z=y';z'=y''\xz'-9z=x\z=uv;z'=u'v+v'u\xu'v+xv'u-9uv=x\xu'v+u(xv'-9v)=x\frac{xdv}{dx}-9v=0|*frac{dx}{vx}\intfrac{dv}{v}=intfrac{9dx}{x}\ln|v|=9ln|x|\v=x^9\frac{x^9du}{dx}=1|*frac{dx}{x^9}\int du=intfrac{dx}{x^9}\u=-frac{1}{8x^8}+C\z=-frac{x}{8}+C_1x^9=y'\y=-frac{x^2}{16}+frac{C_1x^{10}}{10}+C_2$