Differential Equations Answers

According to Newton’s law of cooling, the rate at which a substance cools in moving

air is proportional to the difference between the temperature of the substance and that

of the air. If the temperature of the air is C

o

290 and the substance cools from

C

o

370 to C

o

330 in 10 minutes, find when the temperature will be C

o

295

Solve the ordinary differential equation

d2y/dx2+dy/dx+y=0

Let y1 and y2 be two solutions of the equation a2(x)y'' + a1(x)y' + a0(x)y =0. If W(y1, y2) is the Wronskian of y1 and y2, show that. a2(x)(dw/dx) + a1(x)W = 0

A mass weighing 39.5 kg. stretches a spring 1/4m. At t=0 , the mass is released from a point 3/4m below the equilibrium position with an upward velocity of (5/4)m/second. Determine the function x(t) that describes the subsequent free motion.

Solve the differential equation:

xdy - (3y + x^5. y^1/3)dx = 0

According to Newton’s law of cooling, the rate at which a substance cools in moving air is proportional to the difference between the temperature of the substance and that of the air. If the temperature of the air is 290°C and the substance cools from 370°C to 330°C in 10 minutes, find when the temperature will be 295°C.

iii) The p.d.e.

auxx + 2b uxy + c uyy = 0

where a, b ,c are constants is irreducible when

b 2− ac =0 .

The respiratory flow of air in the lungs is affected due to air pollution. If you have to model

respiratory flow write four essentials for the model.

The rate of increase of susceptible AIDS victims is proportional to the number of susceptible

persons and number of infected persons. If there are 0

S susceptible persons and 1 infected

person at a time to then i) set up the equation for the spread of the disease ii) solve the resulting

equation iii) give a physical interpretation to the same by plotting the epidemic curve iv) write the

limitations of the model.

(a)Calculate all the four second-order partial derivatives of the following functions:

(i) f(x,y)=cos(x^2+y^2)

(ii) f(x,y)=sin(x/y)


(b) Find the range of the function f defined by f(x,y) 10–x^2–y^2 for all (x,y)

for which x^2+y^2 ≤9 Sketch two of its level curves.

(c)Check whether the following functions are differentiable at the point given

against them:

(i)f(x,y)=|x=1|at (1,0)

(ii) f(x,y) =y^3+ ysin2x +e^(x+y) at (1,–1)


(d)Find dw/dt

w = xy + x,z = cos t, y = sint , z=1 at t= 0 .