First- and second-order PDEs, separation of variables, and classic equations of mathematical physics.
These practice problems introduce partial differential equations, where the unknown function depends on more than one independent variable. You will solve first-order equations along characteristics, classify a second-order equation, separate variables on a standard domain, and apply that separation to the heat, wave, and Laplace equations. This is the most advanced differential-equation topic on the site. Solutions keep the ordinary differential equations that fall out of a separation.
First-Order PDEs covers linear and quasilinear equations and the characteristic curves that carry the data. Second-Order Classification asks you to label an equation as hyperbolic, parabolic, or elliptic and to reduce it to a simpler form. Separation of Variables is the main solving method for boundary-value problems on an interval or a rectangle. Equations of Mathematical Physics applies that method to the classical equations, including the boundary conditions that pick the coefficients.
Find the partial differential equation whose general solution is given implicitly by
\[ \Phi\!\left(\frac{z}{x^3},\;\frac{y}{x}\right) = 0, \]
where $\Phi$ is an arbitrary (differentiable) function of two arguments.
Find the partial differential equation whose general solution is given implicitly by
\[ \Phi\!\left(\frac{z}{x^2},\; x - y\right) = 0, \]
where $\Phi$ is an arbitrary differentiable function of two arguments.
Find the partial differential equation whose general solution is
\[ z = (x+y)\,\Phi\!\left(x^2 - y^2\right), \]
where $\Phi$ is an arbitrary differentiable function of one argument.
Find the partial differential equation whose general solution is
\[ z = \tfrac{1}{2}(a^2+2)\,x^2 + axy + bx + \Phi(y+ax), \]
where $a$ and $b$ are arbitrary constants and $\Phi$ is an arbitrary differentiable function.
Find the partial differential equation whose general solution is
\[ z = x^a f\!\left(\frac{y}{x}\right) + x^b \varphi\!\left(\frac{y}{x}\right), \qquad a \neq b, \]
where $f$ and $\varphi$ are arbitrary differentiable functions.
Find the partial differential equation whose general solution is
\[ z = x^a f\!\left(\frac{y}{x}\right) + x^b \varphi\!\left(\frac{y}{x}\right) + x^c \psi\!\left(\frac{y}{x}\right), \]
where $f$, $\varphi$, $\psi$ are arbitrary differentiable functions and $a$, $b$, $c$ are
three distinct constants.
Find the general integral of the equation
\[ s + yq + z = 0, \qquad s = \frac{\partial^2 z}{\partial x\,\partial y},\quad q = \frac{\partial z}{\partial y}. \]
Find the general integral of the equation
\[ y^2 s + 2yp = 1, \qquad s = \frac{\partial^2 z}{\partial x\,\partial y},\quad p = \frac{\partial z}{\partial x}. \]
Find the general integral of the equation
\[ 2yt - xs + 2q = x^2 y, \qquad t = \frac{\partial^2 z}{\partial y^2},\quad
s = \frac{\partial^2 z}{\partial x\,\partial y},\quad q = \frac{\partial z}{\partial y}. \]
Find the general integral of the equation
\[ ps - qr = 0, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y},\quad
r = \frac{\partial^2 z}{\partial x^2},\quad s = \frac{\partial^2 z}{\partial x\,\partial y}. \]
Find the general integral of the equation
\[ pq = zs, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y},\quad
s = \frac{\partial^2 z}{\partial x\,\partial y}. \]
Find the general solution of
\[ xz\,\frac{\partial z}{\partial x} + yz\,\frac{\partial z}{\partial y} = -xy, \]
and then find the integral surface passing through the curve $y = x^2$, $z = x^3$.
Find the general solution of
\[ ap+bq+cz=0, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
where $a$, $b$, $c$ are nonzero constants.
Find the general solution of
\[ x\,\frac{\partial z}{\partial x} + y\,\frac{\partial z}{\partial y}
= z + a\sqrt{x^2+y^2+z^2}, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
where $a$ is a real constant.
Find the general solution of
\[ (y+z)\,\frac{\partial u}{\partial x}
+(z+x)\,\frac{\partial u}{\partial y}
+(x+y)\,\frac{\partial u}{\partial z}
= u, \]
where $u = u(x,y,z)$.
Find the general solution of
\[ xy\,\frac{\partial u}{\partial x}
- \sqrt{1-y^2}\!\left(y\,\frac{\partial u}{\partial y} - z\,\frac{\partial u}{\partial z}\right)
= axy\,\frac{\partial u}{\partial z}, \]
where $u = u(x,y,z)$ and $a$ is a real constant.
Find the general solution of
\[ (s-x)\,\frac{\partial u}{\partial x} +(s-y)\,\frac{\partial u}{\partial y} +(s-z)\,\frac{\partial u}{\partial z} = s - u, \qquad s = x+y+z+u. \]
Find the general solution of
\[ \frac{\partial z}{\partial x} - a\,\frac{\partial z}{\partial y} = e^{mx}\cos py, \]
where $a$, $m$, $p$ are real constants.
Find the general solution of
\[ \frac{\partial u}{\partial x} + b\,\frac{\partial u}{\partial y} + c\,\frac{\partial u}{\partial z} = xyz, \]
where $b$ and $c$ are real constants.
Find the general solution of
\[ (y-bz)\,\frac{\partial z}{\partial x} - (x-az)\,\frac{\partial z}{\partial y} = bx - ay, \]
where $a$ and $b$ are real constants.
Find the integral surface of
\[ y^2\,p + xy\,q = x, \qquad p=\frac{\partial z}{\partial x},\quad q=\frac{\partial z}{\partial y}, \]
that passes through the curve $x=0$, $z=y^2$.
Find the integral surface of
\[ x\,p - 2y\,q = x^2 - y^2, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $y=1$, $z=x^2$.
Find the integral surface of
\[ \tan x\cdot p + y\,q = z, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $y=x$, $z=x^3$.
Find the integral surface of
\[ x\,p + y\,q = z - x^2 - y^2, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $y=-2$, $z=x-x^2$.
Find the integral surface of
\[ xy\,p - y^2\,q = x, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $x=a$, $2ayz=a^2+2$.
Find the integral surface of
\[ yz\,p + xz\,q = xy, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $x=a$, $y^2+z^2=a^2$.
Find the integral surface of
\[ (y-z)\,p + (z-x)\,q = x-y, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $z=y=-x$.
Find the integral surface of
\[ x\,p + (xz+y)\,q = z, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $x+y=2z$, $x-y=1$.
Find the integral surface of
\[ x\,p + z\,q = y, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $y=2z$, $x+2y=z$.
Find the integral surface of
\[ 2xz\,p + 2yz\,q = z^2-x^2-y^2, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $x+y+z=0$, $x^2+y^2+z^2=a^2$.
Find the integral surface of
\[ (y+2z^2)\,p - 2x^2z\,q = x^2, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $x=z$, $y=x^2$.
Find the integral surface of
\[ (x-z)\,p + (y-z)\,q = 2z, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $x-y=2$, $z+2x=1$.
Find the integral surface of
\[ xy^3\,p + x^2z^2\,q = y^3z, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $x=-z^3$, $y=z^2$.
Find the integral surface of
\[ x(x^2+y^2)\,p + 2y^2(xp+yq-z) = 0, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $z=h$, $x^2+y^2=a^2$.
Find the integral surface of
\[ x\,p + y\,q = 2xy, \qquad p = \frac{\partial z}{\partial x},\quad q = \frac{\partial z}{\partial y}, \]
that passes through the curve $y=x$, $z=x^2$.
Find the surface that passes through the line $y=x$, $z=h$ and is
orthogonal to the family of surfaces
\[ x^2+y^2+z^2 = \alpha x, \]
where $\alpha$ is an arbitrary parameter.
Find the surface that passes through the circle $z=h$, $x^2+y^2=a^2$
and is orthogonal to the family of surfaces
\[ xy = \alpha z, \]
where $\alpha$ is an arbitrary parameter.
Find the surface that is orthogonal to the family
\[ x^2+y^2+z^2 = cx, \]
where $c$ is an arbitrary parameter, and passes through the line $y=x$, $z=1$.
Write the partial differential equation satisfied by cylindrical
surfaces whose generators are parallel to the vector $(1,1,2)$, and find the
general solution.
Find the equation of the cylindrical surface whose generators are
parallel to the vector $(1,1,2)$ and whose directrix is the curve
\[ x+y+z=0, \qquad 5x^2+6xy+5y^2=4. \]
Find the equation of the surface whose arbitrary tangent plane
intersects the $Ox$ axis at a point whose abscissa is half the abscissa of the
point of tangency.
Given the partial differential equation $xp+yq=0$.
1° Find its general integral.
2° Among the functions satisfying the given equation, find those that also
satisfy $p^2+q^2=\dfrac{a^2}{x^2+y^2}$.
Let $M=(x_0,y_0,z_0)$ be a point on a surface, $P=(x_0,y_0,0)$ its
orthogonal projection onto $z=0$, and $N$ the intersection of the normal at $M$
with the plane $z=0$. Let $O$ be the origin.
1° Find the general form of surfaces such that $\angle NOP = \pi/4$.
2° Find the surface passing through the $Ox$ axis.
3° Which quadric surfaces have this property?
Given the one-parameter family of quadric surfaces
\[ ax^2 + by^2 - \lambda z^2 = c, \]
where $\lambda$ is the parameter and $a,b,c$ are real constants. Find the
orthogonal trajectories, the one passing through $x=1$, $z^2=y^a$, and determine
when the trajectories are quadric surfaces.
Given the equation
\[ xy^2\,p + xy^2\,q = z(x^2+y^2), \qquad p=\frac{\partial z}{\partial x},\quad q=\frac{\partial z}{\partial y}. \]
1° Find the general integral and the particular integral satisfying
$x=1$, $z=ay(1-y^2)$.
2° Find the general form of the surface satisfying the equation for which
one family of asymptotic lines consists of characteristics.
Given the partial differential equation
\[ (z-y)\,p - (z+x)\,q = x+y. \]
1° Find the general integral.
2° Find the general form of quadric surfaces satisfying the equation.
3° Find the Cauchy integral with $y=x+1$, $z=0$.
4° Investigate the surface from 3° and reduce it to canonical form.
Let $f$ be twice continuously differentiable. Consider
\[ \omega = \left(x\,\frac{\partial u}{\partial x}+y\right)dx
+ \left[x\,\frac{\partial u}{\partial y}+x-2xy+2yf'(u-x)\right]dy. \]
1° Form and integrate the PDE that $u=u(x,y)$ must satisfy for $\omega$ to be a
total differential.
2° Find the integral satisfying $u=y^2$ when $x=0$. For the resulting
$u(x,y)$, find the function whose total differential is $\omega$.
For $\left(\dfrac{p}{x}\right)^2+\left(\dfrac{q}{y}\right)^2=\dfrac{2}{z^2}$:
find the complete and general integrals, and the Cauchy integral through $z=x-1$
at $y=0$.
For $pq=z^mxy$ ($m$ an integer): find the complete and general
integrals; determine for which $m$ the complete integral is a quadric; and for
$m=0$, find the Cauchy integral through $x=1$, $z=\sqrt{1+y^2}$.
For $2xp+2yq+4zp^2-z=0$: find the complete integral (the
general and singular integrals turn out not to exist as real surfaces), and
find the solution satisfying $z^2=x$ on $y=x$.
Find all surfaces $z=\varphi(x,y)$ for which the projection of
$\overrightarrow{OM}$ onto the (upward) unit normal equals
$\dfrac{z-f(p,q)}{\sqrt{1+p^2+q^2}}$, with $f$ homogeneous of degree $n\neq1$;
and for $f$ an entire homogeneous quadratic, find the surface through the
parabola $x=0$, $2z=y^2$.
Solve $\theta_{tt}=a^2\theta_{xx}$ with $\theta(0,t)=\theta(l,t)=0$,
$\theta_t(x,0)=0$, and $\theta(x,0)$ a triangular (tent) profile of peak
$\theta_0$ at $x=l/2$.
For $u_{xx}=u_y\ (*)$: show $u\,dx+u_x\,dy$ is exact and its
potential $v$ also solves $(*)$; verify on an example; and find a solution
of the form $u=Y(y)e^{2ax^2/y}$.
For $\Delta^2u=u_{xxxx}+2u_{xxyy}+u_{yyyy}=0$: verify a list
of polynomial and trig-hyperbolic solutions; and show that if $\varphi,\psi$
are harmonic then $\varphi+x\psi$, $\varphi+y\psi$, and $(x^2+y^2)\varphi$
are biharmonic.