Unformatted text preview: 1. Using the Laplace transform, solve x + x + 5 x = g(t); x(0) = 0, x(0) = 0; g(t) = ¨˙4 ˙ sin t, if 0 ≤ t < π ; 0 if t ≥ π . 2. Consider the oscillator excited by a unit impulse at t = 5, modelled by 2¨ + x + 2x = δ (t − 5), with x˙ x(0) = 0, x(0) = 0. Suppose that it is desired to bring the system to rest again after exactly one cycle, that ˙ is, when the response ﬁrst returns to equilibrium moving in the positive direction. (a) Determine the impulse kδ (t − t0 ) that should be applied to the system in order to accomplish this objective. Note that k is the magnitude of the impulse and t0 is the time of its application. (b) Solve the resulting initial value problem and sketch its solution to conﬁrm that it behaves in the speciﬁed manner. 3. Solve the following diﬀerential equation by means of a power series about x = 0. Find the recurrence relation; also ﬁnd the ﬁrst four terms in each of two linearly independent solutions. y′′ − xy′ − y = 0. 4. The equation y′′ − 2xy′ + λy = 0, −∞ < x < ∞, where λ is a constant, is known as the Hermite equation. (a) Find the ﬁrst four terms in each of two linearly independent solutions about x = 0. (b) Observe that if λ is a nonnegative even integer, then one or the other of the series solutions terminates and becomes a polynomial . Find the polynomial solutions for λ = 0, 2, 4, 6, 8, 10. Note that each polynomial is determined only up to a multiplicative constant. (c) The Hermite polynomial Hn (x) is deﬁnes as the polynomial solution of the Hermite equation with λ = 2n for which the coeﬃcient of xn is 2n . Find H0 (x), . . . , H5(x). 5. Find all values of α for which all solutions of x2 y′′ + αxy′ + (5/2)y = 0 approach zero as x → 0. 1 ...
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- Spring '09
- Laplace, Hermite polynomial Hn, Hermite equation