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Math 302 Exam 1 - Differential Equations - Prof. Jason Metcalfe, Assignments of Differential Equations

University of North Carolina (UNC) - Chapel HillDifferential Equations
Prof. Jason Metcalfe

The practice exercises for exam 1 of math 302 - differential equations, held during summer 2002. The exercises cover various types of differential equations, including first-order and higher-order equations, and require finding general solutions, particular solutions, and classifying equilibrium solutions.

Typology: Assignments

Pre 2010

Uploaded on 03/10/2009

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Math 302 - Differential Equations (Metcalfe)
Summer 2002
June 4, 2002
Exam 1 - Practice Exercises
1. Find a general solution to
dy
dx +xy =e−x2/2
2. Find a solution of:
(3x4y−1) dx +x5dy = 0; y(1) = 1
3. Find a general (implicit) solution of
xy3dx +ex2dy = 0
4. Find a solution of:
y0=xexp(y−x2); y(0) = 0
5. Newton’s law of cooling states that the temperature of an object changes at a rate proportional to
the difference between its temperature and that of its surroundings. Suppose that the temperature
of a cup of coffee obeys Newton’s law of cooling. If the coffee has a temperature of 200◦F when
freshly poured, and 1 minute later has cooled to 190◦F in a room at 70◦F, determine when the
coffee reaches a temperature of 150◦F.
6. Examine the autonomous equation
dy
dt =y3−7y2+ 14y−8
. Find all equilibrium solutions and classify them as stable or unstable.
1
pf3
pf4

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Math 302 - Differential Equations (Metcalfe)

Summer 2002 June 4, 2002

Exam 1 - Practice Exercises

  1. Find a general solution to dy dx + xy = e−x (^2) / 2
  2. Find a solution of: (3x^4 y − 1) dx + x^5 dy = 0; y(1) = 1
  3. Find a general (implicit) solution of

xy^3 dx + ex 2 dy = 0

  1. Find a solution of: y′^ = x exp(y − x^2 ); y(0) = 0
  2. Newton’s law of cooling states that the temperature of an object changes at a rate proportional to the difference between its temperature and that of its surroundings. Suppose that the temperature of a cup of coffee obeys Newton’s law of cooling. If the coffee has a temperature of 200◦^ F when freshly poured, and 1 minute later has cooled to 190◦^ F in a room at 70◦^ F, determine when the coffee reaches a temperature of 150◦^ F.
  3. Examine the autonomous equation

dy dt = y^3 − 7 y^2 + 14y − 8

. Find all equilibrium solutions and classify them as stable or unstable.

  1. Find a general (implicit) solution to

2 xy dx + (y^2 + x^2 ) dy = 0

  1. Find a general (implicit) solution to

(xy^2 + x − 2 y + 3) dx + (x^2 y − 2 x − 2 y) dy = 0

  1. Find a general solution of the equation

y(x + y + 1) dx + (x + 2y) dy = 0

  1. Given that y 1 = x^2 is a solution of

x^2 y′′^ − 3 xy′^ + 4y = 0

Use reduction of order to find the general solution on the interval (0, ∞). Show that the two solutions are independent.

  1. Find a general solution of y′′^ + 2y′^ = 0
  2. Find a solution to y′′^ + 4y′^ + 4y = 0; y(0) = 1; y′(0) = − 1
  3. Use Euler’s formula to prove that

sin(A + B) = sin A cos B + cos A sin B

  1. Find a general solution to y(4)^ + y′′′^ − 4 y′′^ − 4 y′^ = 0
  1. Given that y 1 = x^5 and y 2 = (^) x^1 form a fundamental set of solutions of x^2 y′′^ − 3 xy′^ − 5 y = 0 on (0, ∞). Find the general solution of:

x^2 y′′^ − 3 xy′^ − 5 y =

x^2

using variation of parameters.

  1. Find the general solution of

y′′′^ − 5 y′′^ + 6y′^ = 2 sin x + 8

using the method of undetermined coefficients.