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# 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.

### Contributors

• Integrated by Justin Marshall.