Forming Differential Equations from Their Given Integrals

Find the differential equation of the family of lines $y=ax$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of lines $x^{2}+y^{2}=a^{2}$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of lines $x^{2}+y^{2}=ax$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of lines $y=ax^{2}$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of lines $y=ax+a^{2}$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of lines $\left(x^{2}+y^{2}\right)^{2}=a^{2}\left(x^{2}-y^{2}\right)$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of lines $(x-a)^{2}+y^{2}=1$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of lines $y=ae^{\frac{x}{a}}$ and, where it is simple, note the property of the lines expressed by this equation.

Find the differential equation of the family of confocal second-order curves

where $\lambda$ is a parameter.

Find the differential equation of the family of cycloids $x=a(t-\sin t)$, $y=a(1-\cos t)$, where $a$ is a parameter.

Find the differential equation of the family of cycloids $x+C=a(t-\sin t)$, $y=a(1-\cos t)$, where $C$ is a parameter.

Find the differential equation of the family of tractrices

where $C$ is a parameter.

Find the differential equation of the family of lines with several parameters $y=ax+b$ and, where it is simple, note the property expressed by this equation.

Find the differential equation of the family of lines with several parameters $y=ax^{2}+bx+c$ and, where it is simple, note the property expressed by this equation.

Find the differential equation of the family of lines with several parameters $(x-a)^{2}+(y-b)^{2}=1$ and, where it is simple, note the property expressed by this equation.

Find the differential equation of the family of lines with several parameters $(x-a)^{2}+(y-b)^{2}=c^{2}$ and, where it is simple, note the property expressed by this equation.

Find the differential equation of the family of lines with several parameters $y=A\sin (x+\alpha )$ and, where it is simple, note the property expressed by this equation.

Find the differential equation of the family of lines with several parameters $y=e^{x}(Ax+B)$ and, where it is simple, note the property expressed by this equation.

Prove that the differential equation of all parabolas

has the form

Prove that the differential equation of second-order curves has the form $\left(\left(y''\right)^{-\frac{2}{3}}\right)'''=0$.

Find the system of differential equations satisfied by the coordinates $(x,y,z)$ of points of the family of lines $y=ax$, $z=bx$.

Find the system of differential equations satisfied by the coordinates $(x,y,z)$ of points of the family of lines $x^{2}+y^{2}+z^{2}=a^{2}$, $x+y+z=b$.

Find the system of differential equations satisfied by the coordinates $(x,y,z)$ of points of the family of lines $y=a\cos x$, $z=a\sin x$.

Find the system of differential equations satisfied by the coordinates $(x,y,z)$ of points of the family of lines $x=a\cos t$, $y=a\sin t$, $z=bt$.

Find the differential equations of the normals to the cone $x^{2}+y^{2}=z^{2}$.

Find the differential equations of the normals to the paraboloid $x^{2}+y^{2}=2z$.