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By Reinhard Diestel

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Differentiating, and trusting that the chain rule holds as usual (which it does, as will be proved in Chapter 8), we find ux = ϕ (x − t) + 3 2 and ut = −ϕ (x − t) + = 1 + H(x − t), 3 2 = 2 − H(x − t), uxx = δ(x − t) utt = δ(x − t). Thus, uxx = utt as distributions, and u can be considered as a worthy solution of the wave equation. 25 Find a simpler expression for χ(t)δa (t), where χ is a C 2 function. 26 Determine the derivatives of order ≤ 2 of the functions f (t) = e−|t| , g(t) = |t|e−|t| and h(t) = | sin t |.

A Here we used that the left-hand member is real and thus equal to its own real part. 20 2. 7 Compute the derivative of f (t) = eit by separating into real and imaginary parts. Compare the result with that obtained by using the chain rule, as if everything were real. 8 Show that the chain rule holds for the expression f (g(t)), where g is realvalued and f is complex-valued, and t is a real variable. 9 Compute the integral π eint dt, −π where n is an arbitrary integer (positive, negative, or zero).

Y (3) (t) − y (t) + 4y (t) − 4y(t) = −3et + 4e2t , y(0) = 0, y (0) = 5, y (0) = 3. 23 Solve the system  x (t) − y(t) = e−t , x(0) = 3, x (0) = −2, y(0) = 0. 24 Solve the system  x (t) + 2y (t) + x(t) = 0, x(0) = x (0) = y(0) = 0. 25 Solve the problem y (t) − 3y (t) + 2y(t) = 1, t > 2 ; 0, t < 2 y(0) = 1, y (0) = 0. 26 Solve the system  dy   = 2z − 2y + e−t dt t > 0;   dz = y − 3z y(0) = 1, z(0) = 2. 27 Solve the differential equation 2y (iv) + y − y − y − y = t + 2, t > 0, with initial conditions y(0) = y (0) = 0, y (0) = y (0) = 1.

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