y`/y=(-sinxy)*(xy)`
y`/y=(-sinxy)*(x`*y+x*y`)
x`=1, так как х - независимая переменная
y`/y=(-sinxy)*(y+x*y`)
(y`/y)+x*y`sin(xy)=-y*sin(xy)
y`=-y*sinxy/((1/y)+xsin(xy))
y`=-y^2sin(xy)/(1+(xy)*sin(xy))
б)
(x^2)`*y^2+x^2*(y^2)`+(1/sin^2y)*y`=0
(x^2)`=2x
(y^2)`=(2y)*y`
2x*y^2+(4x^2y+(1/sin^2y))*y`=0
y`=-2xy^2/(4x^2*y+(1/sin^2x))
3.
см. приложение
f(0)=-1
f`(x)=(x)`*arcsinx+x*(arcsinx)` - (sqrt(1-x^2))`
f`(x)=arcsinx + (x/sqrt(1-x^2))+(2x/2sqrt(1-x^2))
f`(x)=arcsinx + 2(x/sqrt(1-x^2))
f`(0)=0
f``(x)=1/sqrt(1-x^2) +(2x)`*(1-x^2)^(-1/2)+2x*((1-x^2)^(-1/2))`;
f``(x)=(1/sqrt(1-x^2))+2*(1-x^2)^(-1/2)+2x*(-1/2)*(1-x^2)^(-3/2)*(1-x^2)`
f``(x)=(1/sqrt(1-x^2))*(3-3x^2+2x)/(1-x^2)
f``(0)=3
По формуле Тейлора
f(x) ≈ -1+0*x+(3/2!)x^2
f(x) ≈ (3/2)x^2 -1
4
dy=y`(x)dx
y`(x)=-sin(arctg(x/2))*(arctg(x/2))`
y`(x)=-sin(arctg(x/2))*(1/(1+(x/2)^2))*(x/2)`
y`(x)=-(1/2)*sin(arctg(x/2))/(1+(x/2)^2)
dy=-(1/2)*sin(arctg(x/2))dx/(1+(x/2)^2)