Question

Даю 100 балов за все заденея кроме 5!!!!!

Answer 1

Решение во вложении...

Answer 2

$1); ; 6xy=(x^2-5)y'\\frac{dy}{dx}=frac{6xy}{x^2-5}; ; ,; ; int frac{dy}{3y}=int frac{2x, dx}{x^2-5}\\frac{1}{3}cdot ln|y|=ln|x^2-5|+C; ,\\ lnBig |frac{x^2-5}{sqrt[3]{y}}Big |+C=0\\\2); ; y'cdot cosx-y, sinx=0; ,; ; y(pi )=-1\\frac{dy}{dx}=frac{y, sinx}{cosx}; ; ,; ; int frac{dy}{y}=int frac{sinx, dx}{cosx}; ; ,\\ln|y|=-ln|cosx|+ln|C|; ,; ; ; ; ln(ycdot cosx)=lnC\\ycdot cosx=C; ; to ; ; y=frac{C}{cosx}\\b); ; -1=frac{C}{cospi }; ; to C=1\\y=frac{1}{cosx}$

$3); ; y^{frac{3}{5}}, dy=cosx, dx\\int y^{frac{3}{5}}, dy=int cosx, dx; ; to ; ; frac{5, y^{frac{8}{5}}}{8}=sinx+C_1; ; ,\\y^{frac{8}{5}}=frac{8}{5}, sinx+C; ; ,; ; C=frac{8}{5}C_1\\y=sqrt[8]{(frac{8}{5}, sinx+C)^5}$

$4); ; y'-xy^2=3xy; ; ,; ; y(0)=2\\frac{dy}{dx}=xy(y+3); ; ,; ; int frac{dy}{y(y+3)}=int x, dx; ,\\int frac{dy}{y(y+3)}=frac{1}{3}int frac{dy}{y}-frac{1}{3}int frac{dy}{y+3}=frac{1}{3}cdot ln|y|-frac{1}{3}cdot ln|y+3|+C; ;\\frac{1}{3}cdot lnBig |frac{y}{y+3}Big |=frac{x^2}{2}+C\\b); ; y(0)=2:; ; frac{1}{3}cdot lnfrac{2}{5}=0+C; ,; ; C=frac{1}{3}, lnfrac{2}{5}\\frac{1}{3}lnBig |frac{y}{y+3}big |=frac{x^2}{2}+frac{1}{3}, lnfrac{2}{5}$