Homogeneous Equations and Those Reducible to Them

Solve the equation $(x+y)\, dx-(x-y)\, dy=0$.

Solve the equation $(x^{2}+2xy-y^{2})\, dx+(y^{2}+2xy-x^{2})\, dy=0$, singling out the integral curve passing through the point $M(1,-1)$.

Solve the equation $x\, dy-y\, dx=y\, dy$.

Solve the equation $y^{2}\, dx+(x^{2}-xy)\, dy=0$.

Solve the equation $(x^{2}+xy+y^{2})\, dx=x^{2}\, dy$.

Solve the equation $(3x^{2}+6xy+3y^{2})\, dx+(2x^{2}+3xy)\, dy=0$.

Solve the equation $\dfrac {dx}{x^{2}-xy+y^{2}}=\dfrac {dy}{2y^{2}-xy}$.

Solve the equation $\dfrac {dy}{dx}=\dfrac {2xy}{3x^{2}-y^{2}}$.

Solve the equation $y'=\dfrac {x}{y}+\dfrac {y}{x}$, singling out the integral curve passing through the point $M(-1,0)$.

Solve the equation $xy'=y+\sqrt{y^{2}-x^{2}}$.

Solve the equation $y\, dx+(2\sqrt{xy}-x)\, dy=0$.

Solve the equation $xy'=y\ln \dfrac {y}{x}$.

Solve the equation $y'(y'+y)=x(x+y)$, singling out the integral curve passing through the point $M(0,0)$.

Solve the equation $(xy'+y)^{2}=y^{2}y'$.

Solve the equation $x^{2}y'^{2}-3xyy'+2y^{2}=0$.

Solve the equation $(x+y+1)\, dx+(2x+2y-1)\, dy=0$.

Solve the equation $(x-2y+5)\, dx+(2x-y+4)\, dy=0$.

Prove that the integral curves of the equation

are logarithmic spirals.

Prove the theorem: if $P\, dx+Q\, dy=0$ is a homogeneous equation, and the expression $P\, dx+Q\, dy$ is a total differential, then the general integral of the given equation has the form $Px+Qy=C$, provided the degrees of $P$ and $Q$ are different from $-1$.

Find the curves for which the triangle between $Oy$, the tangent, and the radius vector from the origin to the point of tangency is isosceles (three variants).

Solve the equation $y(1+xy)\, dx+(1-xy)x\, dy=0$.

Solve the equation $(x^{2}y^{2}+1)y\, dx+(x^{2}y^{2}-1)x\, dy=0$.

Solve the equation $(x^{2}-y^{4})y'-xy=0$.

Solve the equation $\left(y+y\sqrt{x^{2}y^{4}-1}\right)dx+2x\, dy=0$.

Solve the equation $(y^{4}-3x^{2})\, dy+xy\, dx=0$.

Solve the equation $y^{3}\, dx+2(x^{2}-xy^{2})\, dy=0$.

Solve the equation $(x^{2}y^{2}-1)y'+2xy^{3}=0$.

Solve the equation $(x^{6}-y^{4})\, dy=3x^{5}y\, dx$.

Solve the equation $2y'+y^{2}+\dfrac {1}{x^{2}}=0$.

Solve the equation $x^{2}(y'+y^{2})=a(xy-1)$.

Solve the equation $xy^{2}(xy'+y)=a^{2}$.