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Mathematics LibreTexts

2.E: Equations of First Order (Exercises)

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.

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