MATH 308 EXAM 1 REVIEW
(1) Solve the differential equation.
dy
(a) x
= sin x − 2y,
y(π) = 0
dx
(b) 2y 00 = y 0 + 3y
(c) y
dy
+ t = ty 2 ,
dt
y(0) = 2
1
2
(d) 5y 4 ln x − ex cos y + 3
(e)
dy
y5
= ex sin y −
+2
dx
x
2
dy
+ xy 2 = y 2 ,
y(0) = −
dx
3
Solve the equation. Determine the interval in which the solution is defined. Find the maximum value
of the solution in this interval.
3
(f) t2
dy
= (t − t2 )y + t4
dt
(g) (sin y)
cos y
dy
= 4x2 +
,
dx
x
y(1) = 0
4
(h)
dy
ln y + y = xy,
dx
(i) 5y 00 + 9y 0 = 2y,
y(1) = e
y(0) = 2, y 0 (0) = 3
5
(j)
2y
+x
ey
dy
= −1
dx
(2) What is the order of the differential equation? Is it linear or nonlinear?
sin t
(a) t3 y 00 =
y
(b) (ln t)y 0 + yy 0 = sin t
(c) (tan t)y 000 + t5 y + et = 0
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(3) Without solving the equation, find the interval in which the equation has a unique solution.
dy
(a) ln(t − 1) + y = tan t, y(3) = 5
dt
(b) (t + 1)(t + 2)y 00 + 4y 0 + (cot t)y = sin t,
y(−3) = 0, y 0 (−3) = 1
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dy
∂f
= f (t, y) with y(t0 ) = y0 . A theorem states: if f and
are both
dt
∂y
continuous in a region containing the point (t0 , y0 ), then in some interval on the t-axis, the initial value
problem has a unique solution. Find the region in the ty-plane for which the hypotheses of this theorem
are satisfied.√
y
dy
= 3
, y(−2) = 3
(a)
dt
t +1
(4) Consider the initial value problem
(b)
1
dy
= y 3 tan t,
dt
y(2) = −1
8
dy
(5) Consider the equation
= y 3 − 2y 2 − 3y. Find the equilibrium solutions, and classify each one as
dt
asymptotically stable or unstable. Draw the phase line, and sketch several graphs of solutions in the
ty-plane.
(6) Find the Wronskian of two solutions of the differential equation without solving the equation.
t2 y 00 − t(t + 2)y 0 + (t + 2)y = 0
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(7) Consider the equation y 00 − 2y 0 + y = 0. Verify that y1 and y2 are solutions. Find the Wronskian W (y1 , y2 ).
Do they constitute a fundamental set of solutions?
(a) y1 = et , y2 = 2et
(b) y1 = et , y2 = tet
(c) y1 = et + tet , y2 = et − tet
(8) Assume the differential equation
(t2 + 1)y 00 + ty 0 + 5y = 0
has y1 and y2 as a fundamental set of solutions. Let W (y1 , y2 ) denote the Wronskian. If W (y1 , y2 )(0) = 3,
find the value of W (y1 , y2 )(2).
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(9) A tank originally contains 200 gal of water with 20 lb of salt in solution. Water containing 0.5 lb of salt
per gallon is poured into the tank at a rate of 3 gal/min, and the mixture is allowed to flow out of the
tank at the same rate.
(a) Set up the differential equation for the amount Q(t) of the salt in the tank at any time t.
(b) Solve the equation to find Q(t).
(c) Find the time when the concentration of the salt in the tank reaches twice of its original value.
(10) A home buyer wishes to borrow $300,000 at an annual interest rate of 5% to finance the purchase. Assume
the interest is compounded continuously and that payments are also made continuously.
(a) Determine the monthly payment that is required to pay off the loan in 30 years.
(b) Find the total interest paid during the term of the loan.
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(11) A ball with mass 0.2 kg is thrown upward with initial velocity 30 m/s. Assume there is a force due to air
resistance of magnitude |v|/6 directed opposite to the velocity. The acceleration due to the gravity is the
constant g. How long does it take for the ball to reach its maximum height?
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(12) A rocket is launched straight up from the surface of the earth with initial velocity v0 . The gravitational
force acting on the rocket is inversely proportional to the square of the distance from the center of the
Earth. Let R be the radius of the Earth, and g be the acceleration due to gravity at sea level. Neglect air
resistance.
(a) Assume the mass of the rocket is m. Find the gravitational force w(x) acting on the rocket when the
distance from the surface of the Earth to the rocket is x.
(b) Set up a differential equation for the velocity v(x).
(c) Solve the equation.
(d) Determine the maximum altitude that the rocket can reach.
(e) Find the least initial velocity for which the rocket will not return to the Earth.