Higher-Order Equations Admitting a Reduction of Order

Solve the equation $y''=x+\sin x$.

Solve the equation $y'''\sin^{4}x=\sin 2x$.

Solve the equation $y^{IV}=x$.

Find the general integral of the equation $y^{IV}=f(x)$, where $f(x)=x$ for $0<x<1$ and $f(x)=0$ outside this interval.

Find the integral of the equation $y^{IV}=f(x)$, where $f(x)=\left|x\right|$ for $\left|x\right|<1$ and $f(x)=0$ for $\left|x\right|>1$, such that at $x=0$ the quantities $y$, $y'$, $y''$, $y'''$ equal, respectively, $0$, $0$, $0$, $0$.

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

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

Solve the equation $y''^{2}=y'$.

Solve the equation $ay'''=y''$.

Solve the equation $y'''+y''^{2}=0$.

Solve the equation $4y'+y''^{2}=4xy''$.

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

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

Solve the equation $1+y'^{2}+xy'y''=ay''\sqrt{1+y'^{2}}$.

Solve the equation $y'(1+y'^{2})=ay''$.

Solve the equation $y'''=y''^{3}$.

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

Solve the equation $yy''^{2}=1$.

Solve the equation $y''=ae^{y}$.

Solve the equation $3y''=y^{-\frac{5}{3}}$.

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

Solve the equation $1+y'^{2}=2yy''$.

Solve the equation $y^{4}-y^{3}y''=1$.

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

Solve the equation $y''(1+yy')=y'(1+y'^{2})$.

Solve the equation $yy''=y'^{2}$.

Solve the equation $2yy''+y'^{2}+y'^{4}=0$.

Solve the equation $2yy''-3y'^{2}=4y^{2}$.

From the general integral of the equation $yy''+y'^{2}=1$, single out the integral curve passing through the point $(0,1)$, tangent there to the line $x+y=1$.

In the integral of the equation $yy'y''=y'^{3}+y''^{2}$, single out the integral curve passing through the point $(0,0)$, tangent there to the line $x+y=0$.

Find the integral of the equation $y^{3}y''+1=0$ satisfying the initial data $y=1$, $y'=0$ at $x=1$.

Single out the integral curve of the equation $y'^{2}+2yy''=0$ tangent to the line $y=x$ at the point $(1,1)$.

Find the integral curve of the equation $2yy''-3y'^{2}=4y^{2}$ tangent to the line $y=1$ at the point $(0,1)$.

Find the integral of the equation $yy''+n^{2}y^{2}-ky'^{2}=0$ satisfying the initial data $y=a$, $y'=0$ at $x=0$, where $0<k<1$.

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

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

Solve the equation $x^{4}y''+(xy'-y)^{3}=0$.

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

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

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

Solve the equation $x^{2}(yy''-y'^{2})+xyy'=y\sqrt{x^{2}y'^{2}+y^{2}}$.

Solve the equation $xyy''+xy'^{2}-yy'=0$.

Find the integral of the equation

introducing a new function by the substitution $y=x^{2}u$.

Prove that the equation $f(x^{2}y'',xy',y)=0$, where $f$ is a homogeneous function of its arguments, can be reduced to quadratures. (Bourlet.)

Prove that the equation $x^{3}y''=F\left(\frac{y}{x}\right)f(xy'-y)$ can be reduced to quadratures.

Prove that the equation $y''=(1+y'^{2})^{\frac{3}{2}}F\left(\frac{xy'-y}{\sqrt{1+y'^{2}}}\right)f(x^{2}+y^{2})$ can be reduced to quadratures.

Integrate the equation of conic sections

Find the general integral of the equation $xyy''+xy'^{2}+ayy'+f(x)=0$.

Find the general integral of the equation $y''+ay'^{2}+b\sin y=0$.

Find the general integral of the equation $y''+f(y)y'^{2}+\varphi (x)y'=0$.

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

Find the integral from the initial data of the equation $y''y^{2}=-a^{2}$; $x=0$, $y=R$, $y'=0$.

Find the integral from the initial data of the equation $(1+y^{2})y''+2yy'^{2}=y'$; $x=1$, $y=0$, $y'=1$.

Find the integral from the initial data of the equation $(x+a)y''+xy'^{2}=y'$; $x=0$, $y=0$, $y'=1$.

Find the function $u=f(x^{2}+y^{2})$ satisfying Laplace's equation $\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}=0$.

Find the general form of the functions $u=f(x^{2}+y^{2}+z^{2})$ satisfying the three-variable Laplace equation $\dfrac {\partial^{2}u}{\partial x^{2}}+\dfrac {\partial^{2}u}{\partial y^{2}}+\dfrac {\partial^{2}u}{\partial z^{2}}=0$.

Find the general form of the functions $u=f\left(\sqrt{x^{2}+y^{2}}\right)$ satisfying the biharmonic equation

Surfaces for which the mean curvature is equal to zero are called minimal. The coordinates of their points satisfy the equation

where

For surfaces of revolution about the axis $Oz$ one must have $z=f\left(\sqrt{x^{2}+y^{2}}\right)$. Find the minimal surfaces of revolution. (Monge.)

Find the plane curves for which the radius of curvature is proportional to the normal. Consider the cases where the proportionality coefficient $\mu$ equals $\pm 1$ and $\pm 2$.

Find the plane curves for which the radius of curvature is proportional to the cube of the normal.

Find the curves for which the radius of curvature is proportional to the radius vector from the origin to the point of the curve.

For which curves, other than circles, is the radius of curvature equal to the radius vector from the origin?

Determine the equilibrium shape of an inextensible thread acted on by a load such that the load per unit of horizontal projection is the same (the chains of suspension bridges).

Find the equilibrium shape of a homogeneous inextensible thread under the action of its own weight. (Catenary.)

Find the curves for which the radius of curvature is a given function $f(\alpha )=a$ of the angle $\alpha$ formed by the tangent with the axis $Ox$.

Find the curves for which the radius of curvature is a given function $f(\alpha )$ of the angle $\alpha$, where $f(\alpha )\sin^{3}\alpha =a$.

Find the curves for which the radius of curvature is a given function $f(\alpha )$ of the angle $\alpha$, where $f(\alpha )\cos^{2}\alpha =a$.

Find the curves for which the radius of curvature is a given function $f(\alpha )=ae^{m\alpha }$ of the angle $\alpha$ formed by the tangent with the axis $Ox$.

Find the curves for which the radius of curvature is a given function $f(\alpha )=a\alpha$ of the angle $\alpha$ formed by the tangent with the axis $Ox$.

Find the curves for which the radius of curvature is a given function $f(\alpha )=a\sin \alpha$ of the angle $\alpha$ formed by the tangent with the axis $Ox$.

Find the curves for which the radius of curvature is a given function $f(\alpha )=a\sin m\alpha$ of the angle $\alpha$ formed by the tangent with the axis $Ox$.

Find the curves for which the radius of curvature is a given function $f(\alpha )$ of the angle $\alpha$, where $f(\alpha )(1-e^{2}\cos^{2}\alpha )^{\frac{3}{2}}=p$; $0<e<1$.

Find the curves for which the arc length $s$ is a given function of the angle $\alpha$ formed by the tangent at the end of the arc with the axis $Ox$: $s=a\alpha$.

Find the curves for which the arc length $s$ is a given function of the angle $\alpha$, where $s=a\tg \alpha$.

Find the curves for which the arc length $s$ is a given function of the angle $\alpha$, where $s=ae^{m\alpha }$.

Find the curves for which the arc length $s$ is a given function of the angle $\alpha$, where $2s=a\alpha^{2}$.

Find the curves for which the arc length $s$ is a given function of the angle $\alpha$, where $s=a\cos \alpha$.

Find the curves for which the arc length $s$ is a given function of the angle $\alpha$, where $s=a\cos m\alpha$.

Find the curves for which the radius of curvature $R$ is a given function of the arc length $s$: $s^{2}=a(R-a)$.

Find the curves for which the radius of curvature $R$ is a given function of the arc length $s$: $R=ms$.

Find the curves for which the radius of curvature $R$ is a given function of the arc length $s$: $R^{2}+s^{2}=a^{2}$.

Find the curves for which the radius of curvature $R$ is a given function of the arc length $s$: $R^{2}=2as$.

Find the curves for which the radius of curvature $R$ of the curve itself and $R_{1}$ of its evolute are related by $R^{2}+R_{1}^{2}=a^{2}$.

Find the curves for which the radius of curvature $R$ of the curve itself and $R_{1}$ of its evolute are related by $R_{1}^{2}=2aR$.

Find the curves for which the radius of curvature $R$ of the curve itself and $R_{1}$ of its evolute are related by $\dfrac {R^{2}}{a^{2}}+\dfrac {R_{1}^{2}}{b^{2}}=1$.

Find the curve whose arc, measured from some initial point, is equal to the distance of the tangent at the end of the arc from the origin.