1)
y ′ = c s c ( 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 = c s c ( v ) − 1 {\displaystyle v'-1=csc(v)-1}
s i n ( v ) d v = d x {\displaystyle sin(v)dv=dx}
∫ s i n ( v ) d v = ∫ d x {\displaystyle \int sin(v)dv=\int dx}
− c o s ( v ) = x + C {\displaystyle -cos(v)=x+C}
− c o s ( x + y ) = x + C {\displaystyle -cos(x+y)=x+C}
y = a r c c o s ( − x + C ) − x {\displaystyle y=arccos(-x+C)-x}
2)
y ′ = c s c ( y x ) + y x {\displaystyle y'=csc({\frac {y}{x}})+{\frac {y}{x}}}
v = y x {\displaystyle v={\frac {y}{x}}}
v ′ x + v = y ′ {\displaystyle v'x+v=y'}
v ′ x + v = c s c ( v ) + v {\displaystyle v'x+v=csc(v)+v}
v ′ x = c s c ( v ) {\displaystyle v'x=csc(v)}
s i n ( v ) d v = d x x {\displaystyle sin(v)dv={\frac {dx}{x}}}
∫ s i n ( v ) d v = ∫ d x x {\displaystyle \int sin(v)dv=\int {\frac {dx}{x}}}
− c o s ( v ) = l n ( x ) + C {\displaystyle -cos(v)=ln(x)+C}
− c o s ( y x ) = l n ( x ) + C {\displaystyle -cos({\frac {y}{x}})=ln(x)+C}
y = x a r c c o s ( − l n ( x ) + C ) {\displaystyle y=xarccos(-ln(x)+C)}
3)
y c o s ( y 2 ) y ′ − s i n ( y 2 ) = 0 {\displaystyle ycos(y^{2})y'-sin(y^{2})=0}
v = s i n ( y 2 ) {\displaystyle v=sin(y^{2})}
v ′ = 2 y
v ′ 2 − v = 0 {\displaystyle {\frac {v'}{2}}-v=0}
v ′ = 2 v {\displaystyle v'=2v}
d v v = 2 d x {\displaystyle {\frac {dv}{v}}=2dx}
∫ d v v = ∫ 2 d x {\displaystyle \int {\frac {dv}{v}}=\int 2dx}
l n ( v ) = 2 x + C {\displaystyle ln(v)=2x+C}
v = C e 2 x {\displaystyle v=Ce^{2x}}
s i n ( y 2 ) = C e 2 x {\displaystyle sin(y^{2})=Ce^{2x}}
y 2 = a r c s i n ( C e 2 x ) {\displaystyle y^{2}=arcsin(Ce^{2x})}
4)
y ′ = y l n ( y ) + y {\displaystyle y'=yln(y)+y}
v = l n ( y ) {\displaystyle v=ln(y)}
v ′ = y ′ y {\displaystyle v'={\frac {y'}{y}}}
v ′ y = y v + y {\displaystyle v'y=yv+y}
v ′ = v + 1 {\displaystyle v'=v+1}
d v v + 1 = d x {\displaystyle {\frac {dv}{v+1}}=dx}
∫ d v v + 1 = ∫ d x {\displaystyle \int {\frac {dv}{v+1}}=\int dx}
l n ( v + 1 ) = x + C {\displaystyle ln(v+1)=x+C}
v + 1 = C e x {\displaystyle v+1=Ce^{x}}
v = C e x − 1 {\displaystyle v=Ce^{x}-1}
l n ( y ) = C e x − 1 {\displaystyle ln(y)=Ce^{x}-1}
y = e C e x − 1 {\displaystyle y=e^{Ce^{x}-1}}
5)
y ′ = ( x 2 + y − 1 ) 2 − 2 x {\displaystyle y'=(x^{2}+y-1)^{2}-2x}
v = x 2 + y − 1 {\displaystyle v=x^{2}+y-1}
v ′ = 2 x + y ′ {\displaystyle v'=2x+y'}
2 x + y ′ = ( x 2 + y − 1 ) 2 {\displaystyle 2x+y'=(x^{2}+y-1)^{2}}
v ′ = v 2 {\displaystyle v'=v^{2}}
d v v 2 = d x {\displaystyle {\frac {dv}{v^{2}}}=dx}
∫ d v v 2 = ∫ d x {\displaystyle \int {\frac {dv}{v^{2}}}=\int dx}
− 1 v = x + C {\displaystyle -{\frac {1}{v}}=x+C}
v = − 1 x + C {\displaystyle v={\frac {-1}{x+C}}}
x 2 + y − 1 = − 1 x + C {\displaystyle x^{2}+y-1={\frac {-1}{x+C}}}
y = − 1 x + C − x 2 + 1 {\displaystyle y={\frac {-1}{x+C}}-x^{2}+1}
6)
y ′ = x 2 y 2 + y x {\displaystyle y'={\frac {x^{2}}{y^{2}}}+{\frac {y}{x}}}
y ′ = v + x v ′ {\displaystyle y'=v+xv'}
v + x v ′ = 1 v 2 + v {\displaystyle v+xv'={\frac {1}{v^{2}}}+v}
v 2 d v = d x x {\displaystyle v^{2}dv={\frac {dx}{x}}}
∫ v 2 d v = ∫ d x x {\displaystyle \int v^{2}dv=\int {\frac {dx}{x}}}
1 3 v 3 = l n ( x ) + C {\displaystyle {\frac {1}{3}}v^{3}=ln(x)+C}
y 3 3 x 3 = l n ( x ) + C {\displaystyle {\frac {y^{3}}{3x^{3}}}=ln(x)+C}
y = ( 3 x 3 ( l n ( x ) + C ) ) 1 3 {\displaystyle y=(3x^{3}(ln(x)+C))^{\frac {1}{3}}}
