Differential Equations Answers

Given the following monotonically transformed utility function faced by the consumer
lnU(X1X2) = ∝lnX1+βlnX2
The price of good X1 is P1 and the price of good X2 is P2.
Construct the corresponding Langregian function
Derive the optimal demand (Marshallian demand) function for X1 and for X2.

(a) Differentiate the following function
y = x^2 e^x
(b) A cosmetic company is planning the introduction of a promotion of a new lipstick line. The marketing research department after test marketing the new line in a carefully selected city found that the demand in the city is approximately given by p = 12e^(-x), where x which should be within this range, 0 ≤ x ≤ 2 were thousand lipsticks sold per week at a price of Kenya shillings. At what price will the weekly revenue be at maximum? What is the maximum weekly revenue

Separate the following partial differential equation into a set of three ODEs by the method of separation of variables :
d^2 u/dt^2 = c^2 [d^2 u/dr^2 + 1/r * du/dr + 1/r^2 * d^2/d(thita)^2 ]

Solve: z(p-q)=z^2 + (x+y^2)

9. a) Find f ( y) so that equation f (y)dx − zx dy − xy ln y dz = 0 is integrable. Also obtain
the corresponding integral using Natani’s method.

b) Find the differential equation of the family of surfaces
phi [z (x + y)^2 , x^2 - y^2]=0

8. a) Solve the differential equation x^3 p^2 + x^2 yp + a^3 = 0 and also obtain its singular solution, if
it exists.

b) The differential equation satisfied by a beam uniformly loaded (W kg /meter) , with one
end fixed and the second end subjected to a tensile force-P, is given by
EI (d^2 y / d x^2) = Py - 1/2 W x^2 ,

where E is the modulus of elasticity and I is the moment of inertia. Show that the elastic
curve for the beam with conditions y = 0 and dy/dx = 0 at x = 0 , is given by

y = W / Pn^2 (1-cosh nx) + W x^2 / 2P , where n^2 = (P/EI)

c) For 0 < x < 5 and t > 0 , solve the one-dimensional heat flow equation d u/d t = 4 d^2 u/ d x^2
satisfying the conditions u(t, 0) = u(t,5) = 0, u(0, x) = x

10. a) Find the surface which intersects the surfaces of the system z (x+y) = c (3z+1)
orthogonally and which passes through the circle x^2 + y^2 = 1, z = 1

b) Show that the complete integral of z = px + qy - 2p - 3q represents all possible planes
through the points (2, 3, 0)

c) Find the values of n for which the equation (n-1)^2 uxx - y^2n uyy = ny^2n-1 uy is
i) parabolic ii) hyperbolic.

7. a) Solve the following equations:

(ii) (D^2 - 2DD' + D'^2) z = tan (y+x)

b) Solve: z (p-q) = z^2 + (x+y^2)

6. Find the integral curves of the following equations:

a) dx / x^2 - y^2 - yz = dy / x^2 - y^2 - zx = dz / z (x - y)

b) dx / x^2 + y^2 = dy / 2xy = dz / z (x + y)

c) Find the integral surface of the partial differential equation
(x - y) y^2 p + (y - x) x^2 q = (x^2 + y^2 ) z
through the curve xz = a^2, y = 0

5. Apply the method of variations of parameters to solve the following differential equations:

a) x^2 y'' + x y' - y = x^2 e^x

b) y'' + a^2 y = cosec ax

c) Solve the equation d^2 y / d x^2 - cot x dy/dx - sin^2 xy = cos x - cos^3 x by changing the independent
variable.