Physical Math I
Study Guide Midterm 1
1 Find the general or particular solution for the following differential equations. Try to leave your answer
in explicit form when possible.
(a)
dy
= 6e2x−y
dx
dy
= 4x3 y − y,
dx
dy
(d) x2
− 2y = 1,
dx
(c)
√
(b) y ′ = 3 xy
y(1) = −3
y(0) = −4
2 Find the general or particular solution for the following differential equations.
(a) (x2 + 4)y ′ + 2xy = x
(b) xy ′ − 3y = x3 ,
y(1) = 10
(c) y ′ = (1 − y) cos x, y(π) = 2
(d) A 400–gal tank initially contains 100 gal of brine containing 50 lb of salt. Brine containing 1 lb of
salt per gallon enters the tank at the rate of 5 gal/s, and the well-mixed brine in the tank flows out
at the rate of 3 gal/s. Find the amount of salt in the tank at time t. How much salt will the tank
contain when it is full of brine?
(e) Same set up as above except now the brine flows out also at 5 gal/s. How long will it be until only
20 lb of salt is left in the tank?
3 Set up the appropriate form of a particular solution yp , but do not determine the values of the coefficients.
(a) y ′′ + 2y ′ = 8x
(d) y ′′ − 2y ′ + 2y = ex sin x
(b) y ′′ − y ′ + 2y = x3 + 1
(e) 4y ′′ + 4y ′ + y = x2 e−x
(c) y ′′ − y ′ − 6y = e3x
(f) y ′′ + 4y ′ + 4y = x2 e−2x
(g) y ′′ + 4y = 5x cos(2x)
4 Use the method of undetermined coefficients to find the general solution to the given inhomogeneous
equations.
(a) y ′′ + 2y ′ = 8x
(b) y ′′ + 2y ′ − 3y = sin x
(c) y ′′ − y ′ − 2y = 6xex
(d) Solve the following initial value problem:
y ′′ + 4y ′ + 4y = 4x,
y(0) = 1,
y ′ (0) = 2
5 Use the method of variation of parameters to find a particular solution to the given equation.
(a) y ′′ + 4y = tan 2x
6
(b) y ′′ − 4y = xex
(c) y ′′ − 2y ′ − 8y = 3e−2x
(a) Suppose a mass-spring-dashpot system has m = 25, c = 10, and k = 226, and that it is set in
motion with initial position x0 = 20 and initial velocity v0 = 41.
• Find the position function x(t) and sketch a graph of x(t).
• Find the pseudoperiod of the oscillation.
(b) For the following mass-spring-dashpot systems, find the position function x(t) and decided if the
motion is overdamped, critically-damped, or underdamped. If it is underdamped, write the position
function in the form x(t) = Ae−rt cos(ωt − α).
(b1 )
m=3
c = 30
k = 63
x0 = 2
v0 = 2
(b2 )
m=1
c=8
k = 16
x0 = 5
v0 = −10
(b3 )
m=2
c = 12
k = 50
x0 = 0
v0 = −8