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Unformatted text preview: Math 3323 — Exam 1 Name & .115 U 683 September 29, 2000 Put boxes around your answers to make them easier to ﬁnd. 1. (20 points) Solve each of these ﬁrst order linear initial value problems. (a) ix+8y=34e9x, y(0)=8. dy_ 32—5 2. (12 points) Consider the differential equation a — 2 . For which
x . one(s) of the following initial conditions does the Fundamental Existence
and Uniqueness Theorem guaranteee a unique solution? Explain why the
theorem doesn’t apply at the other(s). (a) y(3) = 0. M93— No — ——
M‘gﬁwﬂd ﬂ*(3t0§ M é
M (b) Y(0) = 5 3. (15 points) Solve the exact initial value problem
15x4ydx + (2y + 3x5)dy = 0, y(1) = 2. (Note: It is actually exactyou don’t need to verify it. Just solve it.) 4. (18 points) Use Euler’s method with step size h = 0.2 to estimate at X = 1
the value of the solution to the initial value problem %=2x+ 3y, y(0)=10. To help in keeping your work organized, ﬁll in the following chart with the
values at the intermediate ste  s. 5. (15 points) Find the general solution of the following differential
equation, and write y as an explicit function of X. 6. (20 points) The separable differentiable equation — + x2 = x occurs in certain biological problems. (a) Find the general solution of this differentiable equation, with x written
as an explicit function of t. For each of the following initial conditions, determine what happens to the
solution of the initial value problem as t —> + 00 : Does it converge to — 00, 0, 1, or + 00 ? Explain your answer. (Note: These parts of the problem can
be answered without knowing the solution of the dc. from part (a).) (b) x(0) = 0.1 ...
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 Spring '08
 FALCO
 Boundary value problem, Uniqueness Theorem

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