Systems of Differential Equations

Solve the system $\dfrac {dx}{x}=\dfrac {dy}{y}=\dfrac {dz}{z}$.

Solve the system $\dfrac {dx}{x}=\dfrac {dy}{y}=\dfrac {dz}{x+y}$.

Solve the system $\dfrac {dx}{y}=-\dfrac {dy}{x}=\dfrac {dp}{q}=-\dfrac {dq}{p}$.

Solve the system $\dfrac {dx}{x-y}=\dfrac {dy}{x+y}=\dfrac {dz}{z}$.

Solve the system $(z-y)^{2}\, dy=z\, dx$, $(z-y)^{2}\, dz=y\, dx$.

Solve the system $dx=(x^{3}+3xy^{2})\, dt$, $dy=2y^{3}\, dt$, $dz=2y^{2}z\, dt$.

Solve the system $z\, dy=(z-1)\, dx$, $dx=(y-x)\, dz$.

Solve the system $\dfrac {dx}{z-y}=\dfrac {dy}{x-z}=\dfrac {dz}{y-x}$.

Solve the system $\dfrac {dx}{x}=\dfrac {dy}{y}=\dfrac {dz}{z-\sqrt{x^{2}+y^{2}+z^{2}}}$.

Solve the system $\dfrac {dx}{dt}=\dfrac {x-y}{z-t}$, $\dfrac {dy}{dt}=\dfrac {x-y}{z-t}$, $\dfrac {dz}{dt}=x-y+1$.

Solve the system $\dfrac {dx}{x^{2}-y^{2}-z^{2}}=\dfrac {dy}{2xy}=\dfrac {dz}{2xz}$.

Solve the system $\dfrac {dx}{y(x+y)}=\dfrac {-dy}{x(x+y)}=\dfrac {dz}{(x-y)(2x+2y+z)}$.

Solve the system $\dfrac {dx}{x^{2}-yz}=\dfrac {dy}{y^{2}-yz}=\dfrac {dz}{z(x+y)}$; $z=-1$, $y=1$ at $x=0$.

Solve the system $x\, dy-y\, dx=s\, ds$, $dx^{2}+dy^{2}=ds^{2}$, using the remark: if $a^{2}+b^{2}=c^{2}$, one may set $a=c\cos \varphi$, $b=c\sin \varphi$.

Solve the system $dx^{2}+dy^{2}=a^{2}\, dt^{2}$, $(d^{2}x)^{2}+(d^{2}y)^{2}=b^{2}\, dt^{4}$, using the remark: if $a^{2}+b^{2}=c^{2}$, one may set $a=c\cos \varphi$, $b=c\sin \varphi$.

Solve the system $\dfrac {dy}{dx}+3y+z=0$, $\dfrac {dz}{dx}-y+z=0$; $y=z=1$ at $x=0$.

Solve the system $\dfrac {dx}{dt}+7x-y=0$, $\dfrac {dy}{dt}+2x+5y=0$; $x=y=1$ at $t=0$.

Solve the system $\dfrac {dx}{dt}=z+y-x$, $\dfrac {dy}{dt}=z+x-y$, $\dfrac {dz}{dt}=x+y+z$; $x=1$, $y=z=0$ at $t=0$.

Solve the system $\dfrac {dx}{dt}=y+z$, $\dfrac {dy}{dt}=z+x$, $\dfrac {dz}{dt}=x+y$; $x=-1$, $y=1$, $z=0$ at $t=0$.

Solve the system $\dfrac {dy}{dx}+\dfrac {dz}{dx}=2z$, $3\dfrac {dy}{dx}+\dfrac {dz}{dx}=y+9z$.

Solve the system $\dfrac {dx}{dt}=z-y$, $\dfrac {dy}{dt}=z$, $\dfrac {dz}{dt}=z-x$.

Solve the system $\dfrac {dx}{y}=\dfrac {dy}{z}=\dfrac {dz}{x}$.

Solve the system $\dfrac {dx}{dt}=y-z$, $\dfrac {dy}{dt}=z-2x$, $\dfrac {dz}{dt}=2x-y$.

Solve the system $\dfrac {d^{2}x}{dt^{2}}+2m^{2}y=0$, $\dfrac {d^{2}y}{dt^{2}}-2m^{2}x=0$.

Solve the system $\dfrac {d^{2}x}{dt^{2}}=3(y-x-z)$, $\dfrac {d^{2}y}{dt^{2}}=x-y$, $\dfrac {d^{2}z}{dt^{2}}=-z$.

Solve the system $\dfrac {dx}{dt}=-x+y+z$, $\dfrac {dy}{dt}=x-y+z$, $\dfrac {dz}{dt}=x+y-z$.

Solve the system $\dfrac {d^{2}x}{dt^{2}}=-x+y+z$, $\dfrac {d^{2}y}{dt^{2}}=x-y+z$, $\dfrac {d^{2}z}{dt^{2}}=x+y-z$.

Solve the system $\dfrac {d^{2}y}{dx^{2}}=3y+4z-3$, $\dfrac {d^{2}z}{dx^{2}}+y+z-5=0$.

Solve the system $\dfrac {dx}{dt}=y$, $\dfrac {dy}{dt}=x+e^{t}+e^{-t}$.

Solve the system $x'+5x-2y=e^{t}$, $y'-x+6y=e^{2t}$.

Solve the system $y'=3z-y$, $z'=z+y+e^{ax}$.

Solve the system $x'=2x+4y+\cos t$, $y'+x+2y=\sin t$.

Solve the system $y''+y'+z''-z=e^{x}$, $y'+2y-z'+z=e^{-x}$.

Solve the system $y'-y+z=3x^{2}$, $z'+4y+2z=2+8x$.

Solve the system $x^{2}y''+xz'+y+z=x+1$, $x^{2}z''+xy'-y-z=-x-1$.

Solve the system $x'+n^{2}y=\cos nt$, $y'+n^{2}x=\sin nt$.

Solve the system $x^{2}y'+z=x^{2}$, $z'+y=x$.

Solve the system $y'+z=1$, $x^{2}z'+2y=x^{2}\ln x$.

Solve the system $tx''+2x'+tx=0$, $ty'+2y-tx'=0$.

Solve the system $tx'=t-2x$, $ty'=t(x+y)+2x-t$.

Solve Liouville's system:

introducing the variables $u=x\cos t+y\sin t$, $v=x\sin t-y\cos t$.

Prove that the integration of Hesse's system:

where $X$, $Y$, $Z$ and $T$ are linear forms in the variables $x$, $y$ and $z$, reduces, by the substitutions

to the integration of a system of linear homogeneous equations.

Integrate, using Hesse's method, the system $\dfrac {dx}{dt}=y+xz$, $\dfrac {dy}{dt}=x+yz$, $\dfrac {dz}{dt}=x+z^{2}$.

Integrate, using Hesse's method, the system $\dfrac {dx}{dt}=x+x(x+y)$, $\dfrac {dy}{dt}=z+y(x+y)$, $\dfrac {dz}{dt}=y+z(x+y)$.

Solve by the symbolic method the system of equations

Solve by the symbolic method the system of equations

Solve by the symbolic method the system of equations

Solve by the symbolic method the system of equations

Find the trajectory of a moving point of mass $m$, if the force acting on it is directed perpendicular to the axis $Oz$ and is proportional to the distance from the point to this axis, given that the equations of motion of a material point of mass $m$ under the action of a force with projections $X$, $Y$, $Z$ on the axes have the form $m\dfrac {d^{2}x}{dt^{2}}=X$, $m\dfrac {d^{2}y}{dt^{2}}=Y$, $m\dfrac {d^{2}z}{dt^{2}}=Z$, where $x$, $y$, $z$ are the coordinates of the moving point.

Prove that if the projections of the force acting on a material point of mass $m$ (equations of motion $m\dfrac {d^{2}x}{dt^{2}}=X$, $m\dfrac {d^{2}y}{dt^{2}}=Y$, $m\dfrac {d^{2}z}{dt^{2}}=Z$) are equal to the derivatives of some function $u$ with respect to the corresponding coordinates: $X=\dfrac {\partial u}{\partial x}$, $Y=\dfrac {\partial u}{\partial y}$, $Z=\dfrac {\partial u}{\partial z}$, then the equality $\dfrac {mv^{2}}{2}=u+C$ holds, where $v$ is the speed of the point, equal to $\sqrt{x'^{2}+y'^{2}+z'^{2}}$.

Prove that if the line of action of a force applied to a material point of mass $m$ (equations of motion $m\dfrac {d^{2}x}{dt^{2}}=X$, $m\dfrac {d^{2}y}{dt^{2}}=Y$, $m\dfrac {d^{2}z}{dt^{2}}=Z$) passes through the axis $Oz$, then the projection of the radius vector onto $xOy$ sweeps out equal areas in equal times.

Determine the motion of a body falling from height $h$ in a medium whose resistance is proportional to $v^{n}$, where $v$ is the speed of the body. The initial velocity $v_{0}$ is inclined to the horizon at an angle $\alpha_{0}$.

If $V$ is the potential of an electric field, and $\overrightarrow {H}$ is the strength of a magnetic field, then the equation of motion of a charged particle in vector form is as follows:

Find the motion of the particle, taking $\operatorname {grad}V$ and $\overrightarrow {H}$ to be constant.

The same problem as the previous one (find the motion of a charged particle in vector form $m\dfrac {d^{2}\overrightarrow {r}}{dt^{2}}=e\operatorname {grad}V+e\overrightarrow {H}\times \dfrac {d\overrightarrow {r}}{dt}$, see problem 4469), but