First-Order Linear Partial Differential Equations

Find the general integral of the equation $\cos y\dfrac {\partial z}{\partial x}+\cos x\dfrac {\partial z}{\partial y}=\cos x\cos y$.

Find the general integral of the equation $xy\dfrac {\partial z}{\partial x}-y^{2}\dfrac {\partial z}{\partial y}+x(1+x^{2})=0$.

Find the general integral of the equation $xz\dfrac {\partial z}{\partial x}+yz\dfrac {\partial z}{\partial y}=x$.

Find the general integral of the equation $x^{2}\dfrac {\partial z}{\partial x}-xy\dfrac {\partial z}{\partial y}+y^{2}=0$.

Find the general integral of the equation $x\dfrac {\partial z}{\partial x}+\left(y-\sqrt{R^{2}-z^{2}}\right)\dfrac {\partial z}{\partial y}=0$.

Find the general integral of the equation $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=xy+z$.

Find the general integral of the equation $y^{3}\dfrac {\partial z}{\partial y}-xy^{2}\dfrac {\partial z}{\partial x}=axz$.

Find the general integral of the equation $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=2xy\sqrt{a^{2}-z^{2}}$.

Find the general integral of the equation $y^{2}\dfrac {\partial z}{\partial x}+xy\dfrac {\partial z}{\partial y}=xz$.

Find the general integral of the equation $(x^{2}+y^{2})\dfrac {\partial z}{\partial x}=y^{2}+z^{2}$.

Find the general integral of the equation $xz^{4}\dfrac {\partial z}{\partial x}+yz^{4}\dfrac {\partial z}{\partial y}=x^{2}y^{2}$.

Find the general integral of the equation $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=z+a\sqrt{x^{2}+y^{2}+z^{2}}$.

Find the general integral of the equation $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=z-x^{2}-y^{2}$.

Find the general integral of the equation $(y^{2}+z^{2}-x^{2})\dfrac {\partial z}{\partial x}-2xy\dfrac {\partial z}{\partial y}+2xz=0$.

Find the general integral of the equation $(xy^{3}-2x^{4})\dfrac {\partial z}{\partial x}+(2y^{4}-x^{3}y)\dfrac {\partial z}{\partial y}=9z(y^{3}-x^{3})$.

Find the general integral of the equation $(x^{3}+3xy^{2})\dfrac {\partial z}{\partial x}+2y^{3}\dfrac {\partial z}{\partial y}-2y^{2}z=0$.

Find the general integral of the equation $(x^{2}-y^{2})\dfrac {\partial z}{\partial x}+xy\dfrac {\partial z}{\partial y}+xyz=0$.

Find the general integral of the equation $(y+x)\dfrac {\partial z}{\partial x}+(y-x)\dfrac {\partial z}{\partial y}=z$.

Find the general integral of the equation $(x+y)\left(1+\dfrac {\partial z}{\partial x}+\dfrac {\partial z}{\partial y}\right)+z\dfrac {\partial z}{\partial x}=0$.

Find the general integral of the equation $(z+y-x)\dfrac {\partial z}{\partial x}+(z+x-y)\dfrac {\partial z}{\partial y}=x+y-z$.

Find the general integral of the equation $(z+x)x\dfrac {\partial z}{\partial x}+(z+y)y\dfrac {\partial z}{\partial y}=z^{2}-xy$.

Find the general integral of the equation $(z+e^{x})\dfrac {\partial z}{\partial x}+(z+e^{y})\dfrac {\partial z}{\partial y}=z^{2}-e^{x+y}$.

Find the general integral of the equation $x\dfrac {\partial u}{\partial x}+(z+u)\dfrac {\partial u}{\partial y}+(y+u)\dfrac {\partial u}{\partial z}=y+z$.

Find the general integral of the equation $\dfrac {\partial u}{\partial x}+\dfrac {\partial u}{\partial y}+\dfrac {\partial u}{\partial z}=xyz$.

Find the general integral of the equation $x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=u+\dfrac {xy}{z}$.

Find the general integral of the equation $x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=x^{2}+2u$.

Find the general integral of the equation $xy\dfrac {\partial u}{\partial x}-\sqrt{1-y^{2}}\left(y\dfrac {\partial u}{\partial y}-z\dfrac {\partial u}{\partial z}\right)=axy\dfrac {\partial u}{\partial z}$.

Find the general integral of the equation $(s-x)\dfrac {\partial u}{\partial x}+(s-y)\dfrac {\partial u}{\partial y}+(s-z)\dfrac {\partial u}{\partial z}=s-u$; $s=x+y+z+u$.

Find the general integral of the equation $xz\dfrac {\partial z}{\partial x}+yz\dfrac {\partial z}{\partial y}=xy$.

Find the general integral of the equation $\dfrac {\partial z}{\partial x}-a\dfrac {\partial z}{\partial y}=e^{mx}\cos py$.

Find the general integral of the equation $\dfrac {\partial u}{\partial x}+b\dfrac {\partial u}{\partial y}+c\dfrac {\partial u}{\partial z}=xyz$.

Find the general integral of the equation $\dfrac {1}{\cos x}\dfrac {\partial z}{\partial x}+a\dfrac {\partial z}{\partial y}=z\ctg y$.

Find the general integral of the equation $x\dfrac {\partial z}{\partial x}+\sqrt{1+y^{2}}\dfrac {\partial z}{\partial y}=xy$.

Find the general integral of the equation $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=2xy\sqrt{a^{2}-z^{2}}$.

Find the general integral of the equation $(y-bz)\dfrac {\partial z}{\partial x}-(x-az)\dfrac {\partial z}{\partial y}=bx-ay$.

Find the partial differential equation for conical surfaces whose generators pass through the origin, given that if $u(x,y,z)=0$ is the equation of a surface, then

is the equation of the tangent plane.

Find the partial differential equation for cylindrical surfaces whose generators are parallel to the vector $\overrightarrow {P}(l,m,n)$.

Integrate the equation of conical surfaces $x\dfrac {\partial u}{\partial x}+y\dfrac {\partial u}{\partial y}+z\dfrac {\partial u}{\partial z}=0$. Using the result obtained, find the equation of the conical surface whose directrix has the equation $x^{2}+y^{2}=ax$, $z=1$.

Integrate the equation of cylindrical surfaces, and in the resulting general integral, determine the form of the arbitrary function so that the surface passes through the circle $x^{2}+y^{2}=a^{2}$, $z=1$.

Homogeneous functions of degree $n$ possess, by definition, the property (in the case of three variables): $f(tx,ty,tz)=t^{n}f(x,y,z)$. By Euler's theorem, they satisfy the equality $x\dfrac {\partial f}{\partial x}+y\dfrac {\partial f}{\partial y}+z\dfrac {\partial f}{\partial z}=nf$. Prove the converse theorem: from this partial differential equation follows the functional property of homogeneous functions.

Find the surface passing through the curve $xy=a^{2}$, $z=h$ and satisfying the equation

Find the surface satisfying the equation

and passing through the curve $x=a$, $2ayz=a^{2}+2$.

Find the surface satisfying the equation $(x^{2}+y^{2})\dfrac {\partial z}{\partial x}+2xy\dfrac {\partial z}{\partial y}=xz$ and passing through the curve $x=a$, $y^{2}+z^{2}=a^{2}$.

Find the surface satisfying the equation $z\dfrac {\partial z}{\partial x}-z\dfrac {\partial z}{\partial y}=y-x$ and passing through the curve $x=1$, $z=y^{2}$.

Find the surface satisfying the equation $2xz\dfrac {\partial z}{\partial x}+2yz\dfrac {\partial z}{\partial y}=z^{2}-x^{2}-y^{2}$ and passing through the circle $x+y+z=0$, $x^{2}+y^{2}+z^{2}=a^{2}$.

Find the surface satisfying the equation

and passing through the circle $z=h$, $x^{2}+y^{2}=a^{2}$.

Find the general equation of the surfaces intersecting the cones $xy=\alpha z^{2}$ at a right angle.

Find the equation of the surfaces orthogonal to the surfaces $xyz=\alpha$.

Find the surface passing through the line $y=x$, $z=h$ and orthogonal to the spherical surfaces $x^{2}+y^{2}+z^{2}=\alpha x$.

Find the surface passing through the circle $z=h$, $x^{2}+y^{2}=a^{2}$ and orthogonal to the hyperbolic paraboloids $xy=\alpha z$.

Find the surface passing through the curve $y=x$, $z=x^{3}$ and satisfying the equation $x\dfrac {\partial z}{\partial x}-y\dfrac {\partial z}{\partial y}=z$.

Find the asymptotic lines of the previous surface (see problem 4521), given that the asymptotic lines of a surface $f(x,y,z)=0$ are determined by the equation $\dfrac {\partial^{2}z}{\partial x^{2}}\, dx^{2}+2\dfrac {\partial^{2}z}{\partial x\partial y}\, dx\, dy+\dfrac {\partial^{2}z}{\partial y^{2}}\, dy^{2}=0$.

Find the solution of the equation $z(x+z)\dfrac {\partial z}{\partial x}-y(y+z)\dfrac {\partial z}{\partial y}=0$ that equals $z=\sqrt{y}$ at $x=1$.

Integrate the equation $x^{n}\dfrac {\partial z}{\partial x}+y^{n}\dfrac {\partial z}{\partial y}=z^{n}$. Find the surface satisfying the equation and passing through the line $z=1$, $x=y\sqrt{2}$. Consider the cases $n=\pm 1$.

Find the lines of curvature of the previous surface (see problem 4524) for $n=-1$, given that the lines of curvature of a surface $z=f(x,y)$ are determined by the equation

where $p$, $q$, $r$, $s$, $t$ are, respectively, the partial derivatives $z'_{x}$, $z'_{y}$, $z''_{xx}$, $z''_{xy}$, $z''_{yy}$.

Find the surfaces satisfying the equation

and prove that they are obtained by the motion of circles.

Find the surface satisfying the equation

and passing through the ellipse $x=2z$, $x^{2}+y^{2}=a^{2}$. Prove that this is a ruled surface, and find the orthogonal trajectories of its generators.

Find the solution of the equation $2xz\dfrac {\partial z}{\partial x}+2yz\dfrac {\partial z}{\partial y}=z^{2}-x^{2}-y^{2}$ that contains the hyperbola $x=a$, $z^{2}-y^{2}=a^{2}$.

Integrate the equation

Find the surfaces satisfying the previous equation (see problem 4529) and passing through the lines:

Having found the solution of the equation $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=2z$ passing through the curve $z=y=2x^{3}$, find the orthogonal trajectories of the projections onto $xOy$ of its asymptotic lines (see problem 4522).

Find the surface satisfying the equation

and containing the ellipse $x=0$, $y^{2}+4z^{2}=4a^{2}$.

For the previous surface (see problem 4532), find the lines of steepest descent, i.e., the orthogonal trajectories of the level lines, or of the sections of the surface by the planes $z=C$.

Find the surface satisfying the equation $xy^{2}\dfrac {\partial z}{\partial x}+x^{2}y\dfrac {\partial z}{\partial y}=z(x^{2}+y^{2})$, given that one of the families of its asymptotic lines consists of characteristic lines.

Find the orthogonal surfaces for the family of surfaces of the previous problem (see problem 4534).

Find the surface passing through the circle $z=0$, $x^{2}+y^{2}=y$ and satisfying the equation

Among the functions satisfying the equation $x\dfrac {\partial z}{\partial x}+y\dfrac {\partial z}{\partial y}=0$, find the one that also satisfies the equation

Find the surfaces for which the tangent plane at any point $M$ intersects the axis $Oz$ at a point equidistant from the origin and from the point $M$.

Find the surfaces for which the tangent plane at a point with coordinate $z$ intersects $Oz$ at a point with coordinate $nz$.

Find the solution of the system

in which $u=z^{2}$ at $x=y=0$.