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10_1_1 - x 5 Plot a trajectory of your choice in...

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MTH 108 Section 01. Spring 10 Exam 1, Show all work. Justify procedures and calculations except when they are obvious as, for example, when you integrate both sides of an equation. When doing so, use complete, grammatically correct sentences. No books, notes, calculators. 1. Find the general solution to the ODE y + a 2 y = A cos( at ) , a > 0 . (1) 2. Characterize each of the following ODEs as separable, linear, exact, or homogeneous. Use more than one characterization if more apply. If none applies, state “none”. DO NOT SOLVE. (a) dy dx = x + y 2 x (b) dy dx = x 2 y + sin x (c) x ( x + y ) dy dx = x 2 + y 2 (d) dy dx = 4 x 3 y - 9 x 2 y +1 3 x 3 - x 4 - 1 (e) dy dx = x ( y 2 + 1) 3. Verify that the function e - x 2 satisfies the equation y + (2 - 4 x 2 ) y = 0 . Find the formula of a second, independent solution. Write the general solution of the ODE. 4. Find the general solution of the equation, xy + ( y ) 2 = 0 . Derivatives are with respect to the variable
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Unformatted text preview: x . 5. Plot a trajectory of your choice in phase-space for the ODE below, where b is a small positive number, for example 1 10 , y ±± + 2 by ± + (1 + b 2 ) y = 0 . 6. Let L be a linear operator that can be applied to vectors y . The null space of L is the set of all vectors y that satisfy the equation Ly = 0 Prove that the null space of L is closed under vector addition and scalar multiplication, that is, (a) if two vectors y 1 and y 2 are in the null space of L , then their sum is in the null space of L as well, (b) the scalar multiple of an element of the null space of L is an element of the null space of L . 7. Solve the initial value problem d 2 y dt 2 + 2 dy dt + 5 y = δ ( t-3) , y (0) = 0 , y ± (0) = 0 ....
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