Geometric applications of first-order ODEs

Find a curve $C$ in $\mathbb {R}^{2}$, passing through the point $(3,2)$, with the following property: Let $L(x_0, y_0)$ be the segment of the tangent line to $C$ at $(x_0, y_0)$ which lies in the first quadrant. Then each point $(x_0, y_0)$ of $C$ is the midpoint of $L(x_0, y_0)$.

{#fig-1 width="50%"}

Find the curve for which the mean ordinate, i.e., the quantity $\dfrac {1}{x}\int_{0}^{x}y\, dx$, is proportional to the last ordinate.

Find the curve for which the abscissa of the centroid of the area bounded by the lines $x=0$, $y=0$, $x=\xi$ and the curve itself is equal to $\frac{3}{4}\xi$.

Find the curves for which the length of the subtangent is constant, equal to $a$. (Leibniz, 1684.)

Find the curves for which the subnormal is everywhere equal to $p$.

Find the curves for which the segment of the tangent between the axes is bisected by the point of tangency.

Find the curves for which the same segment (the segment of the tangent between the axes) is divided by the point of tangency in the ratio $m:n$.

Using polar coordinates, find the curves that intersect all radius vectors at an angle $\alpha$, where $\tg \alpha =a$.

Find the curves for which the angle $\varphi$ from the polar axis to the radius vector at the point of tangency is equal to the angle $\mu$ from the extension of the radius vector to the tangent.

The same question as in the previous problem (the angle $\varphi$ from the polar axis to the radius vector at the point of tangency is equal to the angle $\mu$ from the extension of the radius vector to the tangent), but $\mu =2\varphi$.

Find the curve for which the area of the curvilinear trapezoid between the points $(a,0)$ and $(x,0)$ of the axis $Ox$ and the points $A(a,m)$ and $B(x,y)$ of the curve is equal to the product of the arc $AB$ and $m$.

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).

Find the curve for which the area of the trapezoid formed by the coordinate axes, the tangent, and the ordinate at the point of tangency is equal to $a^{2}$. Single out the curve passing through the point $(a,a)$.

Find the curve for which the segment of the tangent between the point of tangency and the axis $Ox$ is seen from the point $(0,1)$ at an angle of $45°$.

Find the curve for which the tangent forms segments on the axes whose lengths sum to $2a$.

Find the curve for which the length of the tangent segment between the axes is equal to $a$.

Find the curves for which the product of the distances from the tangent to two given points is constant.

Find the curve whose tangent forms with the axes a triangle of area $2a^{2}$.

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 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 curve $r = r(\varphi )$, for which the area $S$ of the sector bounded by this curve and the polar rays $\varphi = 0$ and $\varphi = \alpha$ is computed, for any $\alpha \in (0 ; \pi )$, by the formula:

The function $y = f(x)$ is defined, continuous, and positive for $x > 0$. Let $S(c)$ be the area of the figure bounded by the coordinate axes, the line $x = c$, and the graph of the function $y = f(x)$. Find $f$, if $S(c) = \alpha c f(c)$ for any $c > 0(0 < \alpha \leqslant 1)$.

The length of the arc of the graph of a continuously differentiable function $y = f(x)$ from the point $A(0 ; a)$ to the point $M(x ; f(x))$ is proportional, with coefficient $k$, to the slope of the tangent to the graph at the point $M$. Find the function $f$.

Find the curve for which the length of the arc from the point $M_{0}$ to an arbitrary point $M$ on the curve is proportional to the difference $\left|O M\right|- - \left|O M_{0}\right|, O$ being a given point of the plane.

Find the curve $y = f(x), x \geqslant 0(f(x) > 0$ for $x > 0)$, if it is known that for any $a > 0$ the volume of the solid formed by rotating around the axis $O x$ the figure $0 \leqslant x \leqslant a, 0 \leqslant y \leqslant f(x)$ equals $\lambda \pi a f^{2}(a)$, $0 < \lambda < 1$.

Find all functions $y = y(x)$ for which the subnormal is the same at every point of the graph and equals $p > 0$. Indicate the one of these functions whose graph passes through the point $(p / 2 ; p)$. (The subnormal is the projection onto the axis $O x$ of the segment of the normal from a point on the curve to the point of intersection with the axis $O x$.)

Find all curves for which every normal passes through the point $\left(x_{0} ; y_{0}\right)$.

Find the curve passing through the point $(a ;-1), a > 0$, for which the projection onto the axis $O x$ of the segment of any tangent from the point of tangency to the point of intersection with the axis $O x$ equals $a$.

Find, in the first quadrant, the curve passing through the point $(1 ; 1)$, for which the segment of any of its normals, contained between the coordinate axes, is divided by the point of the curve in the ratio $m : n$, $m \neq n$, measured from the axis $O x$.

Find, in the first quadrant, the curve passing through the point $(1 ; 1)$, for which the segment of any tangent, contained between the coordinate axes, is divided by the point of tangency in the ratio $m: n$, measured from the axis $O x$.

Find all curves for which the segment of any normal, from the point on the curve to the point of intersection with the axis $O x$, has length $a$.

Find all curves for which the segment of any tangent, from the point on the curve to the point of intersection with the abscissa axis, has length $a$.

Find the curves passing through the point $(0 ; 2)$ and having the property: the angle between the ray drawn from the origin to an arbitrary point of the curve, and the axis $O x$, equals the angle between this ray and the tangent to the curve at that point.

Find, in the first quadrant, a curve $y = y(x)$ such that the abscissa of the centroid of the homogeneous figure defined by the inequalities $0 \leqslant x \leqslant \xi , 0 \leqslant y \leqslant y(\xi )$, equals $\xi (n-1) / n, \xi > 0, n > 2$.

The illumination at a point of a curve from a point source equals $k I \frac{\sin \varphi }{r^{2}}$, where $I$ is the luminous intensity of the source, $r$ is the distance from the source to the point, $\varphi$ is the angle between the ray from the source to the point and the tangent to the curve at that point. Find the curve for which the illumination at every point from a source of intensity $I$ would be the same and equal to $E$.