Differential Equations Answers

(i)Find the integral of f(x,y ) x^4+y^2 over the region bounded by y=x , y= 2x and x=2

(ii)Find the surface area of the portion of the paraboloid z=25–x^2–y^2 which lies

above the xy -plane.

(iii)Locate and classify the stationary points of the function

f( x,y ) x^2+y^2–6xy +6x +3y –4

(iv)Check whether the following functions are homogeneous or not.

(a)x/y +3y/2x +sin√(x/y)

(b)x^4 +4x^2 +y^2)x^2


(v)Evaluate f(xy )at a point (x,y)for the function f defined by

f(x,y)=x^5 +10x^3 y^3 +8y^4

Verify that the function f satisfies the requirements of Schwarz’s theorem and

hence evaluate f(x,y)

Obtain the differential equation associated with y= Be^x with A&B being arbitrary constants

Classify the following statements as true or false giving reasons for your answers.  i) General solution of the differential equations x^2(d^2y/dx^2)^(3) +y(dy/dx)^(4) +y^4 =0 must  contain four arbitrary constants.


(ii) The set of real (or complex) solutions of equation y'(x) + P(x)y(x)=Q(x) forms a


real (or complex) vector space.


(iii) sin xd^2y/dx^2 +dy/dx +y=0 ]0,π[ is linear homogeneous equation.


iv) The partial differential equation


x^2 ∂^2z/∂x^2 +2xy ∂^2z/∂x∂y +y^2 ∂^2z/∂y^2 –x^m y^n =0


is a reducible homogeneous equation.


(v) The partial differential equation u ∂u/∂x =e^y + sin x, u= u(x,y) is a quasi-linear


p.d.e.

Classify the following statements as true or false giving reasons for your answers.  i) General solution of the differential equations x^2(d^2y/dx^2)^(3) +y(dy/dx)^(4) +y^4 =0 must  contain four arbitrary constants.


(ii) The set of real (or complex) solutions of equation y'(x) + P(x)y(x)=Q(x) fo

Classify the following statements as true or false giving reasons for your answers.  i) General solution of the differential equations x^2(d^2y/dx^2)^(3) +y(dy/dx)^(4) +y^4 =0 must  contain four arbitrary constants.

(ii) The set of real (or complex) solutions of equation y'(x) + P(x)y(x)=Q(x) forms a

real (or complex) vector space.

(iii) sin xd^2y/dx^2 +dy/dx +y=0 ]0,π[ is linear homogeneous equation.

iv) The partial differential equation

x^2 ∂^2z/∂x^2 +2xy ∂^2z/∂x∂y +y^2 ∂^2z/∂y^2 –x^m y^n =0

is a reducible homogeneous equation.

(v) The partial differential equation u ∂u/∂x =e^y + sin x, u= u(x,y) is a quasi-linear

p.d.e.

Using the method of undetermined coefficients, write the trial solution of the equation d^2y/dx^2+2dy/dx+5y=x e^(–1) cos2x and hence solve it.

1. Show that if z is abse nt from the e quation F (x, y, z, p, q)= 0,Charpit's


method coincides with Jacobi's method.



2. how how to solve, by Jacobi's method, a partial differential equation of


the type f( x, delu/delx, delu/delz ) = g(y, delu/dely, delu/delz )

Consider the following first-order ODE formulations :

d n(t)/dt = a(L -n(t)), n(t0)=n0

Associate the physical meaning to the variables {t, n(t)} and the parameters {a, L} so that the above formulation becomes a mathematical model for population changes.

Newton’s law of cooling assumes that air at room temperature is blown past the cooling body (forced cooling). For cooling in still air (natural cooling) a better modal is to assume that the rate of temperature decrease of the cooling body is directly proportional to the the (5/4)th power of the difference between the temperature u of the body and the temperature s of the surrounding air.

i) Write the law for natural cooling as a differential equation. Is this equation linear?

ii) Solve the equation obtained in i) above assuming that initially, the temperature of the cooling

body was u0.

The population x(t) of a certain city satisfies the logistic law dx/dt = (x÷100)- (x^2 ÷ 10^8)

where t is measured in years. Given that the population of the city is 100000 in 1980, determine the population at any time t >1980 . Also find the population in the year 2000.