Question

Розв’язком якого з рівнянь

Answer 1

$y'-\frac{3y}{x}=x^2 \\ \\ y=uv \\ \\ y'=u'v+uv' \\ \\ u'v+uv'-\frac{3uv}{x}=x^2 \\ \\ u'v+u\cdot (v'-\frac{3v}{x})=x^2 \\ \\ 1) \ v'-\frac{3v}{x}=0 \\ \\ \frac{dv}{dx}=\frac{3v}{x} \\ \\ \frac{dv}{v}=\frac{3 \, dx}{x} \\ \\ \int {\frac{dv}{v}}=3\int {\frac{ \, dx}{x}}\\ \\ \ln{|v|}=3\ln{|x|} \\ \\ \ln{|v|}=\ln{|x|^3} \\ \\ v=x^3$

$2) \ u'\cdot x^3=x^2 \\ \\ \frac{du}{dx}=\frac{1}{x} \\ \\ du = \frac{dx}{x} \\ \\ \int {1} \, du = \int {\frac{dx}{x}} \\ \\ u=\ln{x}+C \\ \\ y=uv=(\ln{|x|}+C)\cdot x^3 = x^3\cdot \ln{|x|}+Cx^3$

$y'-\frac{5y}{x}=-2x^2 \\ \\ y=uv \\ \\ y'=u'v+uv' \\ \\ u'v+uv'-\frac{5uv}{x}=-2x^2 \\ \\ u'v+u\cdot (v'-\frac{5v}{x})=-2x^2 \\ \\ 1) \ v'-\frac{5v}{x}=0 \\ \\ \frac{dv}{dx}=\frac{5v}{x} \\ \\ \frac{dv}{v}=\frac{5dx}{x} \\ \\ \int {\frac{dv}{v}} =5\int {\frac{dx}{x}} \\ \\ \ln{|v|}=5\ln{|x|} \\ \\ \ln{|v|}=\ln{|x|^5} \\ \\ v=x^5$

$2) \ u'\cdot x^5=-2x^2 \\ \\ \frac{du}{dx}=-\frac{2}{x^3} \\ \\ du=-\frac{2 \, dx}{x^3} \\ \\ \int {1} \, du =-2\int {x^{-3}} \, dx\\ \\ u=-2\cdot \frac{x^{-2}}{-2} \\ \\ u=\frac{1}{x^2}+C \\\\ y=uv=(\frac{1}{x^2}+C)\cdot x^5 =x^3+Cx^5$

$C=0, \ y=x^3$

Ответ: 2)