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The solutions to various problems from the final exam of math 220, including finding general solutions to odes, approximating solutions using the euler method, and using laplace transforms to solve differential equations. Topics covered include first order odes, second order odes, and the heat equation.
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MATH 220 Final Exam Summer 2005
(1) Find the general solutions to
(a) y′^ + sin(x)y = sin(x)
(b) y′^ = x^2 y^2.
(2) Find the general solution to
y′^ = y − xy^2.
Hint: Let y = 1/v and derive an equation for v(x).
(3) Use the Euler method with step size h = 0.5 to obtain an approximation to y(1) for the problem
y′^ =
1 + xy
, y(0) = 1.
(4) Solve the following ODEs
(a) y′′^ + 4y′^ + 3y = e^3 x^ + e−^3 x
(b) y′′^ + 2y′^ = 1 + sin(x).
(5) Find the general solution to the ODE
y′′^ + xy′^ − y = x^2.
Hint: Find a solution to the homogeneous problem as a polynomial of low degree. Then find a second solution using reduction of order. You can find a particular solu- tion as another polynomial of low degree.
(6) Consider the following system of ODEs for x(t) and y(t)
x′^ = y + t, y′^ = −x + 1.
(a) Find the general solution.
(b) Solve the initial value problem with x(0) = 0 and y(0) = 1.
(10) Consider the periodic function
f (x) = x, − 1 < x < 1; f (x + 2) = f (x).
(a) Sketch f (x) over the range − 3 < x < 3.
(b) What is the average value of f (x)?
(c) Find the Fourier series for f (x).
