PHYS 326 Problem Set #1
Problem 1 : Show that the following equations either are exact or can be made exact, and solve them:
(i) y(2x 2 y 2 + 1) y ′ + x( y 4 + 1) = 0 ;
(ii) [cos2(x) + y sin(2x)] y ′ + y2 = 0 .
Problem 2 : By finding suitable integrating factors, solve the following equations:
(i) (1 − x 2 )y′ + 2x y = (1 − x 2 )3/2 ;
(ii) y′ − y cot(x) + cosec(x) = 0 ;
(iii) (x + y 3 )y′ = y (treat y as the independent variable).
Problem 3 : Solve ( y − x)
dy
+ 2x + 3y = 0 .
dx
Problem 4 : A mass m is accelerated by a time-varying force α e− β t υ3 , where υ is its velocity. It also
experiences a resistive force ηυ , where η is a constant, owing to its motion through the air. The equation of
motion of the mass is therefore
m dυ = α e− β t υ3 − η υ .
dt
Find an expression for the velocity υ of the mass as a function of time, given that it has an initial velocity υ0 .
dy
[Hint: This equation has the form of Bernoulli equation
+ P(x) y = Q(x) y n . ]
dx
Problem 5 : If u = 1 + tan( y) , calculate
dy
= tan(x)cos( y)[cos( y) + sin( y)] .
dx
d[ ln(u )]
; hence find the general solution of
dy