Equations with Separable Variables

Integrate the equation $(1+y^{2})\, dx+xy\, dy=0$.

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

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

Having found the general integral of the equation $\dfrac {x\, dx}{1+y}=\dfrac {y\, dy}{1+x}$, single out the curve passing through the point:

Having found the general integral of the equation $y'\sin x=y\ln y$, single out the curves passing through the points $M(0,1)$ and $M\left(\frac{\pi }{2},1\right)$.

Having found the general integral of the equation $(1+e^{x})yy'=e^{x}$, single out the curve passing through the point $M(1,1)$.

Having found the general integral of the equation $x\sqrt{1-y^{2}}\, dx+y\sqrt{1-x^{2}}\, dy=0$, single out the curve passing through the point $M(0,1)$.

Having found the general integral of the equation $2\sqrt{y}\, dx=dy$, single out the curve passing through the point $M(0,1)$.

Having found the general integral of the equation $y\ln y\, dx+x\, dy=0$, single out the curve passing through the point $M(1,1)$.

Having found the general integral of the equation $(a^{2}+y^{2})\, dx+2x\sqrt{ax-x^{2}}\, dy=0$, single out the curve passing through the point $M(a,0)$.

Find the general integral of the equation $y'+x^{2}y=x^{2}$; single out from it the solution satisfying the condition:

Integrate the equation $(1-x^{2})y'-xy=axy^{2}$.

Integrate the equation $dy+(xy-xy^{3})\, dx=0$.

Integrate the equation $y=xy'^{2}+y'^{2}$.

Integrate the equation $xyy'^{2}+(x^{2}+y^{2})y'+xy=0$.

Integrate the equation $x(1+y'^{2})=1$.

Integrate the equation $x(1+y'^{2})^{\frac{3}{2}}=a$.

Integrate the equation $x=\dfrac {ay’}{\sqrt{1+y'^{2}}}$.

Integrate the equation $y\sqrt{1+y'^{2}}=y'$.

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 $\frac{3}{4}\xi$.

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

Integrate the equation $y'=\sin (x-y)$.

Integrate the equation $y'=ax+by+c$.

Integrate the equation $y'^{2}=ax+by+c$.

Integrate the equation $y'=(ax+by+c)^{2}$.

Integrate the equation $(x+y)^{2}y'=a^{2}$.

Find the general integral of the equation

applying the substitution

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 $\mu =2\varphi$.

On the surface $z^{2}=2ay$ find the lines whose tangents make a given angle $\gamma$ with $Oz$.

The same question (find the lines whose tangents make a given angle $\gamma$ with $Oz$) for the surface $x^{2}+y^{2}=2az$.

Find the curves that intersect all parallels of the surface of revolution $x^{2}+y^{2}=a^{2}\ch^{2}\frac{z}{a}$ at an angle of $45°$.

The same question (find the curves that intersect all parallels of the surface of revolution at an angle of $45°$) for the cone $x^{2}+y^{2}=z^{2}$.

Find the curves that intersect all generators of the cylinder $y=\varphi (x)$ at an angle $\alpha$.

Find the loxodrome, i.e., the curve intersecting at an angle $\alpha$ all meridians of the sphere $x^{2}+y^{2}+z^{2}=a^{2}$. (Here it is best to introduce spherical coordinates for the points of the sphere's surface.)

What angle with the meridians does the loxodrome make that runs from Izmail on the Danube to Cape Dezhnev? Their latitudes are $45°20'$ and $66°$, and their longitudes are $28°45'$ and $170°$ (east).

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 $m$.

A vessel contains $M$ cm$^{3}$ of solution. Into it, every second, $q$ cm$^{3}$ of water flows continuously and is immediately mixed in. From the vessel, $q$ cm$^{3}$ of solution flows out per second. Prove that the amount of dissolved substance $m$ is expressed by the formula $m=m_{0}e^{-\frac{q}{M}t}$, where $m_{0}$ is the initial amount of substance, and $t$ is the time in seconds from the start of the process.

A rope is wound around a post of radius $r$. The force pressing its element $\Delta s$ against the post is equivalent to the quantity $\frac{T}{r}\Delta s$, where $T$ is the tension of the rope at the given point. Owing to friction, the element can only slip if the force driving it is greater than $kf$, where $k$ is the coefficient of friction and $f$ is the force pressing the element against the post. Derive from this the differential equation $dT=-\frac{kT}{r}\, ds$ and its general integral $T=T_{0}e^{-\frac{ks}{r}}$, where $T$ is the tension at which slipping begins and $T_{0}$ is the tension at the initial point.

Light of brightness $f_{0}$ falls on a light-absorbing medium. Assuming that the absorption by a layer $\Delta x$ is approximately proportional to $\Delta x$ and to the brightness $f$, prove that at depth $x$ the brightness is expressed by the equality $f=f_{0}e^{-kx}$, where $k$ is the coefficient of proportionality.

Rays from a light source are absorbed by the surrounding medium. Assuming that the absorption of light between spheres of radii $r$ and $r+\Delta r$ centered at the light source is, up to small quantities of higher order, equal to $kf\cdot 4\pi r^{2}\Delta r$, prove that the brightness at a distance $r$ from the light source equals $f_{0}\dfrac {e^{-kr}}{r^{2}}$, where $f_{0}e^{-k}$ is the brightness of light at $r=1$, and $k$ is the coefficient of proportionality.

Find the curve that would be equally illuminated everywhere (with a given illuminance $a$) by a point source of light of intensity $A$. It is known that illuminance is inversely proportional to the square of the distance and proportional to the sine of the angle between the ray and the illuminated surface.

Water flows uniformly ($V$ cm$^{3}$/sec) into a cylindrical vessel with base area $S$ cm$^{2}$ and height $H$ cm, and simultaneously partly flows out through an opening in the bottom of area $\sigma$ cm$^{2}$. After what time will the vessel fill to the top, and what is the condition for it to ever fill up? The outflow velocity is determined by Torricelli's law (see problem 3302).

A heavy material point of mass $m$ begins to fall in a resisting medium, with the resistance force depending on the velocity according to the law $f=kv^{n}$. Determine the law of change of velocity over time and the distance traveled in time $t$ for $n=1$; $2$.

A gas of initial density $\rho_{0}$ and pressure $p_{0}$ is contained in a vessel rotating about its axis with constant angular velocity $\omega$. Find the law of pressure distribution in the steady-state motion, when all particles of the gas rotate with angular velocity $\omega$. The gas obeys Boyle-Mariotte's law; neglect the weight of the gas.