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Integrating Factor - Differential Equations - Exam, Exams of Differential Equations

Some keywords in Differential Equations are Convolution, Laplace Transform, Implicit Solution, Initial Condition, Integrating Factor, Autonomous Differential Equation, Appropriate Substitution. Some points of this exam paper are: Integrating Factor, Solve, Solution, Verify, Cooling States, Newton’S Law, Temperature

Typology: Exams

2012/2013

Uploaded on 03/31/2013

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Prof. S. Brick Differential Equations; Exam 1 Math 238
Summer ’05 section 101
Print your name:
Show all of your work, and explain your reasoning. Give the correct units
where appropriate.
1. Find the integrating factor ρ(x) for the ODE: tan(x)y0+ sec2(x)y=√x4+ 1.
Do not solve the ODE.
2. Verify that y=ln(x)
x2is a solution to x2y00 + 5xy0+ 4y= 0.
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Prof. S. Brick Differential Equations; Exam 1 Math 238

Summer ’05 section 101

Print your name:

Show all of your work, and explain your reasoning. Give the correct units where appropriate.

  1. Find the integrating factor ρ(x) for the ODE: tan(x)y′^ + sec^2 (x)y =

x^4 + 1. Do not solve the ODE.

  1. Verify that y =

ln(x) x^2

is a solution to x^2 y′′^ + 5xy′^ + 4y = 0.

  1. Newton’s Law of cooling states that the rate of change of the temperature of an object in a medium of constant temperature is proportional to the difference of its temperature and that of its surroundings. Give a differential equation for this Law, explicitly defining each of your terms.
  2. The function y = tan(x^3 + C) is a solution to the differential equation y′^ = 3x^2 (y^2 + 1) (you need not check that). Find the value of C so that the initial condition y(0) = 1 is satisfied.
  1. Solve y′^ = 3x^2 (y^2 + 1).
  2. Transform the homogeneous equation xy^2 y′^ = x^3 + y^3. into a separable equation. Show it is separable, by separating the variables (but don’t integrate).
  1. Transform the Bernoulli equation xy′^ + 6y = 3xy^4 /^3 into a first order linear equation. Do not solve.
  2. Solve (1 + yexy^ )dx + (2y + xexy^ )dy = 0