Differential Equations Answers

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.

For second order phase transitions, show that

Δp/ΔT ={(Cp2–Cp1)/(Tv(αp2–αp1))}

Plot temperature variation of heat capacity for second order phase transition. How does this plot is different from the plot of lambda transitions

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.

If a simple pendulum of length l oscillates through an angle α on either side of the mean position then find the angular velocity dt/dθ
of the pendulum where θ is the angle which the string makes with the vertical.

A particle of mass m is thrown vertically upward with velocity v0. The air resistance is mg cv^2 where c is a constant and v is the velocity at any time t. Show that the time taken by
the particle to reach the highest point is given by v0 underroot c= tan (gt underroot c )