I'll assume the ODE is
Solve the homogeneous ODE,
The characteristic equation
has roots at and
. Then the characteristic solution is
For nonhomogeneous ODE (1),
consider the ansatz particular solution
Substituting this into (1) gives
For the nonhomogeneous ODE (2),
take the ansatz
Substitute (2) into the ODE to get
Lastly, for the nonhomogeneous ODE (3)
take the ansatz
and solve for .
Then the general solution to the ODE is
Answer:
See explanation
Step-by-step explanation:
Solution:-
- The demand function of a certain brand is given as price P a function of x quantity of goods ( in hundred ) demanded per month. The relation is:
P ( x ) = 50 Ln ( 50 / x ).
- The point price elasticity ( E ) of demand is given by:
- Where, dP / dx : is the rate of change of price ( P ) with each hundred unit of good ( x ) is demanded.
- To determine the " dP / dx " by taking the first derivative of the given relation:
P ( x ) = 50 Ln ( 50 / x ).
d P ( x ) / dx = [ 50*x / 50 ] * [ -1*50 / x^2 ]
= - 50 / x
- Hence the point price elasticity of demand is given by:
E = - ( P / x ) * ( 50 / x )
E = -50*P / x^2
- For an inelastic demand, ! E ! is < 1:
! -50*P / x^2 ! < 1
50*P / x^2 < 1
P < x^2 / 50
- For an unitary demand, ! E ! is = 1:
! -50*P / x^2 ! = 1
50*P / x^2 = 1
P = x^2 / 50
- For an inelastic demand, ! E ! is > 1:
! -50*P / x^2 ! > 1
50*P / x^2 > 1
P > x^2 / 50
2)
If the unit price is increased slightly from $50, will the revenue increase or decrease?
- We see from the calculated demand sensitivity d P / dt:
d P ( x ) / dx = - 50 / x
- We see that as P increases the from P = $50, the quantity of goods demanded would be:
50 = 50 ln(50/x)
1 = Ln ( 50 / x )
50/x = e
x = 50 / e
Then,
d P ( x ) / dx = - 50 / ( 50 / e )
d P ( x ) / dx = - e
- We see that if price slightly increases from $ 50 then the quantity demanded would decrease by e (hundreds ) goods.
- The decrease in the quantity demanded is higher than the increase in price. The revenue is given by the product of price P ( x ) and x:
Revenue R ( x ) = P ( x ) * x
= 50*x*ln(50/x)
Then the product of price and quantity goods also decreases; hence, revenue decreases.
