Classify the following into linear and non-linear models, justifying your classification.
i) Simple harmonic motion for small amplitude of oscillation.
ii) Population growth model given by are constants.
iii) Equation for velocity v of a particle at any time t, moving with a constant acceleration a, and initial velocity u.
iv) Equation describing dynamic stability of market equilibrium price given by are constants and
is the price in period t.
### Classification of Models into Linear and Non-Linear A linear model is one in which the dependent variable and its derivatives appear to the first power and are not multiplied or divided by each other. A non-linear model involves variables raised to powers other than one, multiplied by each other, or appearing in trigonometric, exponential, or other non-linear functions. #### (i) Simple Harmonic Motion for Small Amplitude of Oscillation The equation of simple harmonic motion (SHM) is typically given as: \[ \frac{d^2x}{dt^2} + \omega^2 x = 0 \] where \( x \) is the displacement and \( \omega \) is the angular frequency. - This equation is linear because it involves the second derivative ________ _______ _________ ________ ________.
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Given all the portfolios of n securities what criterion would an investor use to select a good portfolio?
Consider the epidemic model governed by the following equation
with initial condition x = n at t = 0 . Here )t(x is the number of susceptibles at time ,t β is the contact rate. The population is assumed to be closed and homogeneously mixing. Let the contact rate be 0.002 and the number of susceptibles be 5000 initially
i) Find the density of the population when the rate of appearance of new cases is maximum.
ii) Find the time (in weeks) at which the rate of appearance of new cases is maximum.
iii) Obtain the maximum rate of appearance of new cases.
For a given set of securities, all their portfolios lie on or within the boundary of the region shown in Fig.1.
In the feasible region, find a portfolio which has maximum return. Also, find a portfolio in this region which has minimum risk.
A string of length l is connected to a fixed point at one end and to a stick of mass m at the other. The stick is whirling in a circle at constant velocity v . Use dimensional analysis to find the equation of the force in the string.
Consider the group of individuals born in a given year )0 t( = and let )t(n be the number of these individuals surviving t year later. Let )t(x be the number of members of this group who have not had smallpox by year t and are therefore still susceptible. Let β be the rate at which susceptibles contract smallpox and let v be the rate at which people who contract smallpox die from the disease. Finally, let )t( µ be the death rate from all causes other than smallpox. If dx dt/ and dn dt/ are, respectively the rates at which the number of susceptibles and entire population decline due to contraction from smallpox and also due to death from all causes then
i) Formulate the above problem by writing equations for dx dt/ and dn dt/ .
ii) Taking z = x/n , show that z satisfies the initial value problem
iii) Find )t(z at any time t.
iv) Bernoulli estimated that Using these values, determine the proportion of 20 years old who have not had smallpox.
A particle moving in .H.S M has got the velocities cm8 /sec and cm6 /sec when it is at distance cm3 and cm4 respectively from the centre of its motion. Determine the period and the amplitude of motion.
Suppose that the previous forecast was 2090 and the actual value of the variable of interest for the period was 1985 and the oldest value of interest was 1955. Using the moving average technique based upon the most recent four observations find new forecast for the next period.