Inverse Laplace Transform

Find the inverse Laplace transform of the following in the attached file. Thanks.

Transfer Function

A battery of voltage Vi is connected in series with a resistor of resistance R, an inductor of inductance L and a capacitor of capacitance C. If, the output voltage across capacitor is Vo, derive the transfer function.

Second order to first order

Find the equivalent first-order system (that is, find the matrix A and the vector R of dv/dx = Av + R) for the second order equation (see attachment for equation)

Initial value problem

(See attached file for full problem description with proper symbols) --- 1. Solve the initial-value problem d2u/dt2 + w2u = (w2-2)cos(t), u(0)=2, du/dt (0) = 0 where w and  are constants. Show that the solution can be written in the form 2cos(t)cos(t) where =(-w)/2 a ...continues

Linear Differential Equations

Please see attached file for homework specifics. Thank-you for your help. UPDATE: Maple would be satisfactory where HPGSolver is suggested.

Linear Differential Equations

(See attached file for full problem description)

Legendre equation

(See attached file for full problem description)

Euler's equation

Euler's equation states (z^2)w'' + αzw' + βw = 0 where w'=dw/dz If we substitute in w=z^u into Euler's equation, we find the so-called indicial equation: u^2 + (α-1)u +β=0 Use the change of variable t=ln(z) in the Euler equation to show that the Euler equation has the linearly independent pair of solut ...continues

General Solution to the Differential Equation

Help! I am having a problem with this homogenous differential equation (y^2 + yx)dx + x(^2)dy=0 I tried substitution of x = vy, dx = vdy + ydv, but I am not quite getting it right. Please help!

Difficult Diff. EQ

This problem is really giving me issues, as I do not even know how to set it up to solve it. Please solve this problem in as much detail as possible. I guess the number 1 in the parentheses is what is confusing me. Thank you so much! (1 + x^2 + Y^2 +(x^2)(y^2))dy = (y^2)dx