Miscellaneous Exercises

1. Integrate the following equations:
(i) $(1-x^2{)}^{\frac{1}{2}}dx+(1-y^2{)}^{\frac{1}{2}}dy=0$;
(ii) $x(1+y^2{)}^{\frac{1}{2}}dx+y(1+x^2{)}^{\frac{1}{2}}dy=0$;
(iii) $(x^2+2xy-y^2)dx-(x^2-2xy-y^2)dy=0$;
(iv) $(y^2-xy)dx+(x^2-xy)dy=0$;
(v) $x^{2}ydx+(x^3-y^3)dy=0$;
(vi) $(x+y)dx-(2x-y-1)dy=0$;
(vii) $(x+2y+1)dx-(2x+4y+3)dy=0$;
(viii) $(2x^2+6xy+y^2)dx+(3x^2+2xy+4y^2)dy=0$;
(ix) $(x^2+y^2)dx+xydy=0$;
(x) $(1+x^2)p+xy=1$;
(xi) $p+y\tan x=\sin 2x$;
(xii) $p+y\cos x=e^{\sin x}$;
(xiii) $xp-ay=x^n$;
(xiv) $xp-y=x^2\sin x$;
(xv) $p+2xy=x{e}^{-x^2}$;
(xvi) $p\sin x\cos x-y={\sin }^{3}x$;
(xvii) $(y^2-xy)dx+(x+1)dy=0$;
(xviii) $(y-xp{)}^2=4p$;
(xix) $y-px+p(p-1)=0$;
(xx) $xy{p}^2+(x^2+y^2)p+xy=0$;
(xxi) $y{p}^3+2px-y=0$;
(xxii) $y-(x+5)p+p^3=0$;
(xxiii) $(x+1)p^2-(x+y)p+y=0$;
(xxiv) $y-2px+p^2=0$.

2. Determine $n$ so that the equation

$$\frac{a{x}^2+2bxy+c{y}^2}{(x^2+y^2{)}^n}(ydx-xdy)=0$$

is exact.

3. Show that the equation

$$(y^4-2y^2)dx-(3x{y}^3-4xy+y)dy=0$$

has an integrating factor which is a function of $x{y}^2$, and solve the equation.