Question

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

Answer 1

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