1)
y′=csc(x+y)−1{\displaystyle y'=csc(x+y)-1}
v=x+y{\displaystyle v=x+y}
v′=1+y′{\displaystyle v'=1+y'}
v′−1=csc(v)−1{\displaystyle v'-1=csc(v)-1}
sin(v)dv=dx{\displaystyle sin(v)dv=dx}
∫sin(v)dv=∫dx{\displaystyle \int sin(v)dv=\int dx}
−cos(v)=x+C{\displaystyle -cos(v)=x+C}
−cos(x+y)=x+C{\displaystyle -cos(x+y)=x+C}
y=arccos(−x+C)−x{\displaystyle y=arccos(-x+C)-x}
2)
y′=csc(yx)+yx{\displaystyle y'=csc({\frac {y}{x}})+{\frac {y}{x}}}
v=yx{\displaystyle v={\frac {y}{x}}}
v′x+v=y′{\displaystyle v'x+v=y'}
v′x+v=csc(v)+v{\displaystyle v'x+v=csc(v)+v}
v′x=csc(v){\displaystyle v'x=csc(v)}
sin(v)dv=dxx{\displaystyle sin(v)dv={\frac {dx}{x}}}
∫sin(v)dv=∫dxx{\displaystyle \int sin(v)dv=\int {\frac {dx}{x}}}
−cos(v)=ln(x)+C{\displaystyle -cos(v)=ln(x)+C}
−cos(yx)=ln(x)+C{\displaystyle -cos({\frac {y}{x}})=ln(x)+C}
y=xarccos(−ln(x)+C){\displaystyle y=xarccos(-ln(x)+C)}
3)
ycos(y2)y′−sin(y2)=0{\displaystyle ycos(y^{2})y'-sin(y^{2})=0}
v=sin(y2){\displaystyle v=sin(y^{2})}
v′=2yy′cos(y2){\displaystyle v'=2yy'cos(y^{2})}
v′2−v=0{\displaystyle {\frac {v'}{2}}-v=0}
v′=2v{\displaystyle v'=2v}
dvv=2dx{\displaystyle {\frac {dv}{v}}=2dx}
∫dvv=∫2dx{\displaystyle \int {\frac {dv}{v}}=\int 2dx}
ln(v)=2x+C{\displaystyle ln(v)=2x+C}
v=Ce2x{\displaystyle v=Ce^{2x}}
sin(y2)=Ce2x{\displaystyle sin(y^{2})=Ce^{2x}}
y2=arcsin(Ce2x){\displaystyle y^{2}=arcsin(Ce^{2x})}
4)
y′=yln(y)+y{\displaystyle y'=yln(y)+y}
v=ln(y){\displaystyle v=ln(y)}
v′=y′y{\displaystyle v'={\frac {y'}{y}}}
v′y=yv+y{\displaystyle v'y=yv+y}
v′=v+1{\displaystyle v'=v+1}
dvv+1=dx{\displaystyle {\frac {dv}{v+1}}=dx}
∫dvv+1=∫dx{\displaystyle \int {\frac {dv}{v+1}}=\int dx}
ln(v+1)=x+C{\displaystyle ln(v+1)=x+C}
v+1=Cex{\displaystyle v+1=Ce^{x}}
v=Cex−1{\displaystyle v=Ce^{x}-1}
ln(y)=Cex−1{\displaystyle ln(y)=Ce^{x}-1}
y=eCex−1{\displaystyle y=e^{Ce^{x}-1}}
5)
y′=(x2+y−1)2−2x{\displaystyle y'=(x^{2}+y-1)^{2}-2x}
v=x2+y−1{\displaystyle v=x^{2}+y-1}
v′=2x+y′{\displaystyle v'=2x+y'}
2x+y′=(x2+y−1)2{\displaystyle 2x+y'=(x^{2}+y-1)^{2}}
v′=v2{\displaystyle v'=v^{2}}
dvv2=dx{\displaystyle {\frac {dv}{v^{2}}}=dx}
∫dvv2=∫dx{\displaystyle \int {\frac {dv}{v^{2}}}=\int dx}
−1v=x+C{\displaystyle -{\frac {1}{v}}=x+C}
v=−1x+C{\displaystyle v={\frac {-1}{x+C}}}
x2+y−1=−1x+C{\displaystyle x^{2}+y-1={\frac {-1}{x+C}}}
y=−1x+C−x2+1{\displaystyle y={\frac {-1}{x+C}}-x^{2}+1}
6)
y′=x2y2+yx{\displaystyle y'={\frac {x^{2}}{y^{2}}}+{\frac {y}{x}}}
y′=v+xv′{\displaystyle y'=v+xv'}
v+xv′=1v2+v{\displaystyle v+xv'={\frac {1}{v^{2}}}+v}
v2dv=dxx{\displaystyle v^{2}dv={\frac {dx}{x}}}
∫v2dv=∫dxx{\displaystyle \int v^{2}dv=\int {\frac {dx}{x}}}
13v3=ln(x)+C{\displaystyle {\frac {1}{3}}v^{3}=ln(x)+C}
y33x3=ln(x)+C{\displaystyle {\frac {y^{3}}{3x^{3}}}=ln(x)+C}
y=(3x3(ln(x)+C))13{\displaystyle y=(3x^{3}(ln(x)+C))^{\frac {1}{3}}}