Miscellaneous Problems

Find the integral of the equation $(x+y)(1-xy)\, dx+(x+2y)\, dy=0$.

Find the integral of the equation $(x^{2}+xy-ay)\, dx+ax\, dy=0$.

Find the integral of the equation $\left(y+\sqrt{x^{2}+y^{2}}\right)dx=x\, dy$.

Find the integral of the equation $2x\, dx+(x^{2}+y^{2}+2y)\, dy=0$.

Find the integral of the equation $(x-y-4)\, dx=(x+y-2)\, dy$.

Find the integral of the equation $2xy'+y^{2}-1=0$.

Find the integral of the equation $(x^{2}+y^{2})y'=xy$.

Find the integral of the equation $y=xy'+y'^{2}$.

Find the integral of the equation $xy'+2y=\sqrt{y}$.

Find the integral of the equation $xy'-y=y^{2}$.

Find the integral of the equation $(3x^{2}+6xy+y)\, dx+(3x^{2}+x-1)\, dy=0$.

Find the integral of the equation $x(x-1)y'-y=(x-1)^{2}$.

Find the integral of the equation $(y^{2}+2xy-x)y'=y^{2}$.

Find the integral of the equation $(y+xy')^{2}=x^{2}y'$.

Find the integral of the equation $xy'=2y+\sqrt{1+y'^{2}}$.

Find the integral of the equation $y'^{3}-4xyy'+8y^{2}=0$.

Find the integral of the equation $y^{2}(1+y'^{2})=a(x+yy')$.

Find the integral of the equation $2y=x^{2}+xy'+y'^{2}$.

Find the integral of the equation $4y=(x+y')^{2}$.

Find the integral of the equation $4y=x^{2}+y'^{2}$.

Find the integral of the equation $2y=2x^{2}+4xy'+y'^{2}$.

Find the integral of the equation $x^{2}(y-xy')=yy'^{2}$.

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

Find the integral of the equation $y'=m\ln x+\ln (xy'-y)$.

Find the integral of the equation $x^{3}y'^{2}+x^{2}yy'+a^{3}=0$.

Find the integral of the equation $(x^{2}-1)y'+y^{2}-2xy+1=0$.

Find the integral of the equation $y=\left(y'-\frac{1}{2}\sqrt{x}\right)^{3}$.

Find the integral of the equation $(xy'-y)^{2}=y'^{2}-2\frac{y}{x}y'+1$.

Find the integral of the equation $(1+6y^{2}-3x^{2}y)y'=3xy^{2}-x^{2}$.

Find the integral of the equation $y^{2}-(y'^{2}+y'^{3})xy+x^{2}y'^{5}=0$.

Find the integral of the equation $x^{2}+y=y'^{2}$.

Find the integral of the equation $y'\sqrt{a^{2}+x^{2}}+y=\sqrt{a^{2}+x^{2}}-x$.

Find the integral of the equation $y-xy'=a(y^{2}+y')$.

Find the integral of the equation $\left(\sin (x+y)+x\cos (x+y)\right)dx+x\cos (x+y)\, dy=0$.

Find the integral of the equation $(x^{2}+1)y'+x\sin y\cos y-x(x^{2}+1)\cos^{2}y=0$.

Find the integral of the equation $(x^{2}+y^{2}+a)yy'+(x^{2}+y^{2}-a)x=0$.

Find the integral of the equation $(x^{2}y^{2}+1)y+(xy-1)^{2}xy'=0$; find the integral curve passing through the point $M(a,1)$.

Find the integral of the equation $y'^{2}-yy'+e^{x}=0$.

Find the integral of the equation $(x^{2}-a^{2})\, dx+2xy\, dy=0$; find the integral curve passing through the point $M(2a,3a)$.

Find the integral of the equation $y'=ay^{n}+bx^{\frac{n}{1-n}}$.

Find the integral of the equation $(x^{2}-y^{2})\, dx+2xy\, dy=0$; find the integral curve passing through the point $M(a,3a)$.

Find the integral of the equation $(x^{2}-1)\, dx+(x^{2}y^{2}+x^{3}+x)\, dy=0$.

Find the integral of the equation $2\left(x^{2}y+\sqrt{1+x^{4}y^{2}}\right)dx+x^{3}\, dy=0$.

Find the integral of the equation $xyy'=x^{2}y'+y^{2}$; find the integral curve passing through the point $M(1,1)$.

Find the integral of the equation $(x+2y^{3})y'=y$.

Find the integral of the equation $xyy'-y^{2}=(x+y)^{2}e^{-\frac{y}{x}}$.

Find the integral of the equation $y-(x+5)y'-y'^{3}=0$.

Find the integral of the equation $axy'+by=x^{m}y^{n}(\alpha xy'+\beta y)$.

Find the integral of the equation $(x^{n}+y^{2})\, dx-2xy\, dy=0$, $n>1$; find the integral curve passing through the point $M(a,0)$.

What shape must a mirror be given, its surface being a surface of revolution, so that it reflects all rays parallel to the axis into the origin?

Find the curves for which the length of the normal is equal to the distance from the point of tangency to the origin.

Find the curve such that the distances from the origin to the tangent and to the normal at a variable point of the curve are in a constant ratio.

Find the curve for which the product of the segments formed by the tangent and the normal on the axis $Ox$ is equal to $a^{2}$.

Find the curve for which the ratio of the distances from the normal to two given points is constant.

Find the curve for which the subnormal at any point is equal to the radius vector from the origin to that point.

A point moves along the axis $Ox$ and drags a point $M$ connected to it by a rod of length $a$. Find the path of the point $M$ (the tractrix of a line).

The same question as in the previous problem, but the moving point traces out the circle $x^{2}+y^{2}=a^{2}$; the initial position of the point $M$ is on the polar axis.

Find the curve along which, moving without friction, a heavy point would perform uniform horizontal motion.

Find the curve sliding along which a heavy point would descend by equal distances in equal intervals of time.

Prove that the functional equation $f(2x)=2f(x)$ has the solution $f(x)=ax$, unique among functions having a derivative at $x=0$.

Find the curves for which the projection of the ordinate onto the normal is constant.

Find the curves for which the initial ordinate of the tangent is equal to the abscissa of the point of tangency.

Find the curves for which the distance from the tangent to the origin is proportional to the length of the normal.

Find the lines for which the area of the sector bounded by any radius vectors is proportional to the length of the arc bounding that sector.

Find the lines for which the midpoint of the tangent segment between the coordinate axes lies on the parabola $y^{2}=2px$.

Find the curves for which the area of the triangle formed by the axis $Ox$, the radius vector of a point, and the tangent at it, has a constant value $Q$.

Find the lines for which the segment of the axis $Oy$ between the tangent and the normal has a constant length $a$.

Find the curves for which the arc length is related to the coordinates of its endpoint by $s=x-\dfrac {a^{2}}{y}$.

Find the curves for which the arc length $s$ is proportional to the square of the abscissa.

Find the curves for which $s=f(y)$, where $f(y)$ is a given function.

Find the curve for which $s^{2}=8ay$.

Find the curve for which $s^{2}=y^{2}-a^{2}$.

Find the curve for which $y=ae^{\frac{s}{a}}$.

Find the curves for which the centroid of the arc, taken from the origin to any point, has an abscissa equal to $\dfrac {1}{n}$ of the abscissa of the end of the arc.

Find the slopes of the tangents drawn at the point $(1,2)$ to the two integral curves of the equation $y'^{2}=y^{2}-2x+x^{2}$ passing through this point. Show that the radius of curvature of each of these curves at this point is equal to $4$.

Find the locus of points having the property that the two integral curves of the equation $xy'^{2}-yy'+1=0$ passing through these points intersect:

Find the curve that, after a rotation about the origin through any angle $\alpha$ and a corresponding proportional change of both coordinates, would coincide with the curve before the rotation.

Find the plane curve that, after any stretching of the ordinates by a factor $k$, would turn out to be similar to itself, i.e., would coincide with the curve obtained from the given one by a proportional change of both coordinates by a factor $\lambda$.

Find the asymptotic lines of the conoids $z=\varphi \left(\frac{y}{x}\right)$. (For asymptotic lines, the osculating plane coincides with the tangent plane to the surface.)

Find the lines of curvature of the hyperbolic paraboloid $az=xy$. The lines of curvature of a surface satisfy the equation

where

Find the lines of curvature of the ellipsoid $\dfrac {x^{2}}{a^{2}}+\dfrac {y^{2}}{b^{2}}+\dfrac {z^{2}}{c^{2}}=1$. (Monge.)