Differential and Laplace transformations
1. Find an equation of the form y" + p(t)y' + q(t)y = 0 that has solutions y1 = x^4 and y = x^4 (2 +lnx)
2. A circuit has a capacitor .25 x 10^-6 farad, a resistor 5x10^3ohms and an inductor of 1 henry, and an externally applied voltage of 2e^-1000t. The intial charge on the capacitor is 10,000 coulombs and the initial current is 1000 amps. Find q(t) for any time t.
3. Find Laplace transform of t^3e^-4t cos 2t.
4. Find inverse laplace transform of 2s^3 - 2s^2 + s + 1/ s^4 - 3s^3 +6s^2 + 10s
Any hints to these problems would be appreciated.