a) Show that the Cobb-Douglas production function $P=$ $L^\alpha K^\beta$ satisfies the equation
$$L \frac{\partial P}{\partial L}+K \frac{\partial P}{\partial K}=(\alpha+\beta) P .$$
The constants $\alpha$ and $\beta$ are fixed. $L$ is labor and $K$ is capital. b) Verify that $f(x, y)=\sqrt{x^2+y^2}$ satisfies the Eikonal equation $f_x^2+f_y^2=1$. This partial differential equation is important in optics. We will later see that it can be written as $|\nabla f|=1$.

Solution:
\begin{aligned} & P=L^\alpha K^\beta \text {, so } \frac{\partial P}{\partial L}=\alpha L^{\alpha-1} K^\beta \text { and } \frac{\partial P}{\partial K}=\beta L^\alpha K^{\beta-1} \text {. Then } \\ & L \frac{\partial P}{\partial L}+K \frac{\partial P}{\partial K}=L\left(\alpha L^{\alpha-1} K^\beta\right)+K\left(\beta L^\alpha K^{\beta-1}\right)=\alpha L^{1+\alpha-1} K^\beta+ \\ & \beta L^\alpha K^{1+\beta-1}=(\alpha+\beta) L^\alpha K^\beta=(\alpha+\beta) P \text {. } \end{aligned}

1 a) The following functions solve either the Laplace equation $u_{x x}+u_{y y}=0$ or the wave equation $u_{x x}-u_{y y}=0$. Decide in each case. Possible answers are “none”, “both”, “Wave equation” or “Laplace equation”. As usual $\log =\ln$ is the natural log.
a) $u=2 x^2+2 y^2$
b) $u=x^3+3 x y^2$
c) $u=\log \sqrt{x^2+y^2}$
d) $u=e^{-x} \cos y-e^{-y} \cos x$
e) $u=\sin (5 x) \sin (5 y)$
f) $u=\left(\frac{y}{y^2-x^2}\right)$
g) $u=x^4-6 x^2 y^2+y^4$
h) $u=\sin (x-y)+$ $\log \left(x^2+y^2\right)$

(a) Wave equation.
(b) Wave equation
(c) Laplace equation.
(d) Laplace’s Equation.
(e) Wave equation
(f) Wave equation.
(g) Laplace equation.
(h) None

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