2.E: Equations of First Order (Exercises)

These are homework exercises to accompany Miersemann's "Partial Differential Equations" Textmap. This is a textbook targeted for a one semester first course on differential equations, aimed at engineering students. Partial differential equations are differential equations that contains unknown multivariable functions and their partial derivatives. Prerequisite for the course is the basic calculus sequence.

Q2.1

Suppose \(u:\mathbb{R}^2\mapsto\mathbb{R}^1\) is a solution of
$$
a(x,y)u_x+b(x,y)u_y=0 .
$$
Show that for arbitrary \(H\in C^1\) also \(H(u)\) is a solution.

Q2.2

Find a solution \(u\not\equiv const.\) of
$$
u_x+u_y=0
$$
such that
$$
\mbox{graph}(u):=\{(x,y,z)\in\mathbb{R}^3:\ z=u(x,y),\ (x,y)\in\mathbb{R}^2\}
$$
contains the straight line \((0,0,1)+s(1,1,0),\ s\in\mathbb{R}^1\).

Q2.3

Let \(\phi(x,y)\) be a solution of
$$
a_1(x,y)u_x+a_2(x,y)u_y=0\ .
$$
Prove that level curves \(S_C:=\{(x,y):\ \phi(x,y)=C=const.\}\) are characteristic curves, provided that \(\nabla\phi\not=0\) and \((a_1,a_2)\not=(0,0)\).

Q2.4

Prove Proposition 2.2.

Q2.5

Find two different solutions of the initial value problem

\[u_x+u_y=1,\]

where the initial data are \(x_0(s)=s\), \(y_0(s)=s\), \(z_0(s)=s\).

Hint: \((x_0,y_0)\) is a characteristic curve.

Q2.6

Solve the initial value problem
$$
xu_x+yu_y=u
$$
with initial data \(x_0(s)=s,\ y_0(s)=1\), \(z_0(s)\), where \(z_0\) is given.

Q2.7

Solve the initial value problem
$$
-xu_x+yu_y=xu^2,
$$
\(x_0(s)=s,\ y_0(s)=1\), \(z_0(s)=\mbox{e}^{-s}\).

Q2.8

Solve the initial value problem
$$
uu_x+u_y= 1,
$$
$x_0(s)=s,\ y_0(s)=s$, \(z_0(s)=s/2\) if \(0<s<1\).

Q2.9

Solve the initial value problem
$$
uu_x+uu_y= 2,
$$
\(x_0(s)=s,\ y_0(s)=1\), \(z_0(s)=1+s\) if \(0<s<1\).

Q2.10

Solve the initial value problem \(u_x^2+u_y^2=1+x\) with given initial data \(x_0(s)=0,\ y_0(s)=s,\ u_0(s)=1,\
p_0(s)=1,\ q_0(s)=0\), \(-\infty<s<\infty\).

Q2.11

Find the solution \(\Phi(x,y)\) of
$$
(x-y)u_x+2yu_y=3x
$$
such that the surface defined by \(z=\Phi(x,y)\) contains the curve
$$
C:\ \ x_0(s)=s,\ y_0(s)=1,\ z_0(s)=0,\ s\in{\mathbb R}.
\]

Q2.12

Solve the following initial problem of chemical kinetics.
$$
u_x+u_y=\left(k_0e^{-k_1x}+k_2\right)(1-u)^2,\ x>0,\ y>0
$$
with the initial data \(u(x,0)=0,\ u(0,y)=u_0(y)\), where \(u_0\), \(0<u_0<1\), is given.

Q2.13

Solve the Riemann problem
\begin{eqnarray*}
u_{x_1}+u_{x_2}&=&0\\
u(x_1,0)&=&g(x_1)
\end{eqnarray*}
in \(\Omega_1=\{(x_1,x_2)\in\mathbb{R}^2:\ x_1>x_2\}\) and in \(\Omega_2=\{(x_1,x_2)\in\mathbb{R}^2:\ x_1<x_2\}\),
where
$$
g(x_1)=\left\{\begin{array}{r@{\quad:\quad}l}
u_l&x_1<0\\
u_r&x_1>0
\end{array}\right.
$$
with constants \(u_l\not=u_r\).

Q2.14

Determine the opening angle of the Monge cone, that is, the angle between the axis and the apothem (in German: Mantellinie) of the cone, for equation
$$
u_x^2+u_y^2=f(x,y,u),
$$
where \(f>0\).

Q2.15

Solve the initial value problem
$$
u_x^2+u_y^2=1,
$$
where \(x_0(\theta)=a\cos\theta,\ y_0(\theta)=a\sin\theta,\ z_0(\theta)=1,
\ p_0(\theta)=\cos\theta\), \(q_0(\theta)=\sin\theta\) if \(0\le\theta<2\pi\),
\(a=const.>0\).

Q2.16

Show that the integral \(\phi(\alpha,\beta;\theta,r,t)\), see the Kepler problem, is a complete integral.

Q2.17

a) Show that \(S=\sqrt{\alpha}\ x +\sqrt{1-\alpha}\ y +\beta\), \(\alpha,\
\beta\in\mathbb{R}^1, \ 0<\alpha<1\), is a complete integral of \(S_x-\sqrt{1-S_y^2}=0\).
b) Find the envelope of this family of solutions.

Q2.18

Determine the length of the half axis of the ellipse
$$
r=\frac{p}{1-\varepsilon^2\sin(\theta-\theta_0)},\ 0\le\varepsilon<1.
\]

Q2.19

Find the Hamilton function \(H(x,p)\) of the Hamilton-Jacobi-Bellman differential equation if \(h=0\) and \(f=Ax+B\alpha\), where
\(A,\ B\) are constant and real matrices, \(A:\ \mathbb{R}^m\mapsto \mathbb{R}^n\), \(B\) is an orthogonal real \(n\times n\)-Matrix and \(p\in\mathbb{R}^n\) is given. The set of admissible controls is given by
$$
U=\{\alpha\in\mathbb{R}^n:\ \sum_{i=1}^n\alpha_i^2\le1\}\ .
\]

Remark. The Hamilton-Jacobi-Bellman equation is formally the Hamilton-Jacobi equation \(u_t+H(x,\nabla u)=0\), where the Hamilton function is defined by
$$
H(x,p):=\min_{\alpha\in U}\left(f(x,\alpha)\cdot p+h(x,\alpha)\right),
$$
\(f(x,\alpha)\) and \(h(x,\alpha)\) are given. See for example, Evans [5], Chapter 10.

Contributors and Attributions

• Prof. Dr. Erich Miersemann (Universität Leipzig)

• Integrated by Justin Marshall.