Differential Equations 00 4th Edition solutions

Differential Equations 00 4
dy dt = y

Differential Equations 00 4
dy dt = y + 1

Differential Equations 00 4
dy dt = 2t y

Differential Equations 00 4
dy dt = t 2 y + 1

Differential Equations 00 4
dy dt = y + e2t

Differential Equations 00 4
dy dt = 2y + sin t

Differential Equations 00 4
d2 y dt2 + 2y = cost

Differential Equations 00 4
d2 y dt2 + 5 dy dt + y = sin 2t

Differential Equations 00 4
Verify that y(t) = tan t is a solution of dy dt = y2 + 1, and compute a power series solution to find the terms up to degree six (up to and including the t6 term) of the Taylor series centered at t = 0 of tan t.

Differential Equations 00 4
d2 y dt2 + 2y = 0

Differential Equations 00 4
d2 y dt2 + dy dt + y = 0

Differential Equations 00 4
d2 y dt2 + dy dt + t 2 y = cost

Differential Equations 00 4
d2 y dt2 + t dy dt + y = e2t

Differential Equations 00 4
Compute the Hermite polynomials H3(t), H4(t), and H5(t). Check that they are solutions of the appropriate Hermites equation.

Differential Equations 00 4
Legendres equation is the second-order differential equation (1 t 2) d2 y dt2 2t dy dt + ( + 1)y = 0, where is a constant. Guess the Taylor series centered at t = 0 as a solution y(t), that is, y(t) = a0 + a1t + a2t 2 + a3t 3 + a4t 4 + .... (a) Compute the coefficients a2, a3, and a4 in terms of t

Differential Equations 00 4
In this appendix we used the fact that two power series in t are equal if and only if all of their coefficients are equal. To justify this statement, let f (t) = a0 + a1t + a2t 2 + a3t 3 + ... and g(t) = b0 + b1t + b2t 2 + b3t 3 + ... and suppose that f (t) = g(t) for all t for which both sides are d

Differential Equations 00 4
Consider the linear differential equation dy dt = y + et . The natural first guess for a particular solution of this equation is y(t) = aet , but this guess does not yield a solution because it is a solution of the associated homogeneous equation. Guess a power series solution y(t) = a0 + a1t + a2t2

Differential Equations 00 4
Consider the linear differential equation d2 y dt2 + 4y = cos 2t. The natural first guess for a particular solution of this equation is yp(t) = a sin 2t + b cos 2t, but this guess does not yield a solution because it is already a solution of the associated homogeneous equation. Guess a power series y