OldExam_1_Answers001 - Math 3323 — Exam 1 Name&.115 U...

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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 exact--you 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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