MAT 2384 Practice Final Exam
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MAT 2384 Practice Final Exam
Question 1. Solve the initial-value problem:
(4xy - y - 3y2 sin x)dx + (-2x + 4x2 + 9y cos x)dy = 0, y(0) = 1
Question 2. Use the method of undetermined coefficients to find the general solution of the differential equation
y\\\ - y\\ + 2y = -5e-x + 6x + 2.
Question 3. Use variation of parameters to find the general solution of the differential
equation
x y2\\ - 5xy + 5y\ = 4x6 ln x, x > 0.
Question 4. Use Gaussian quadrature of order 3 to estimate the value of the integral
3
e-x2 dx.
1
(Round the values of the coefficients and the nodes to 5 decimal places.)
Question 5. Find the general solution to the following nonhomogeneous system:
d北(d) ┌ ┐y(y)2(1) = ┐ ┌ ┐y(y)2(1) + ┌ -8北(2)北+41┐ .
Question 6. Use the Runge–Kutta method of order 4 to estimate the values of the solution
of
dy
in the interval [0, 1] using a stepsize of h = 0.5. Round your answer to 6 decimal places.
Question 7. (a) Find the Laplace transform of the function f (t) = te-2t cos(5t).
14e-3s (b) Find the inverse Laplace transform of the function F (s) =
Question 8. Use Laplace transforms to solve the initial-value problem
y\\ + 16y = 6(t - 2), y(0) = 2, y\ (0) = -3, where 6(t - 2) is the Dirac delta function.
Question 9. Consider the three points (xj , fj ), j = 0, 1, 2, where fj = f (xj ) for a certain
function f :
(0.3, 0.036591), (0.5, 0.055783), (1, 0.049787).
(a) Find p2 (x), the Lagrange polynomial of order 2 that passes through the three points (round the coefficients to six decimal places).
(b) Interpolate the value of the function f at x = 0.8 (round your answer to six decimal places).
(c) If we know that 0.032145 ● <f\\\ (t)< ● 0.41201 for some t in the interval [0.3, 1], find the range of the error, and use it to determine how accurate your estimate for f (0.8) is.
Question 10. Use a power series to find the general solution to
y\\ - 5y = 0
2022-07-19