Consider the following Harrod-Domar growth equation: \[ \frac{s}{\theta} = g + \delta \] where \( s \) is the saving rate, \( \theta \) is the capital-output ratio, \( g \) is the overall growth rate, and \( \delta \) is the capital depreciation rate. If \( \delta = 0 \) and \( s = 20% \), then to achieve \( g = 10% \), the capital-output ratio will be ________ (in integer).
Let \( Y \) be income, \( r \) be the interest rate, \( G \) be government expenditure, and \( M_s \) be money supply. Consider the following closed economy IS-LM equations with a fixed general price level (\( \bar{P} \)):
IS equation: \[ Y = 490 + 0.6Y - 4r + G \] LM equation: \[ \frac{M_s}{\bar{P}} = 20 + 0.25Y - 10r \] If \( G = 330 \) and \( \frac{M_s}{\bar{P}} = 500 \), then the equilibrium \( Y \) is ________ (round off to one decimal place).
Find the values of \( x, y, z \) if the matrix \( A \) satisfies the equation \( A^T A = I \), where
\[ A = \begin{bmatrix} 0 & 2y & z \\ x & y & -z \\ x & -y & z \end{bmatrix} \]
(b) Order of the differential equation: $ 5x^3 \frac{d^3y}{dx^3} - 3\left(\frac{dy}{dx}\right)^2 + \left(\frac{d^2y}{dx^2}\right)^4 + y = 0 $