Differential Equations Answers

A simple series circuit has an inductor of 1henry , a capacitor of 10^-6 farads and a resistor of 1000ohms . The initial charge on the capacitor is zero. If a 12volt battery is connected to the circuit and the circuit is closed at t=0, find the charge on the capacitor 1sevond later and the steady state charge.

For 0<x<5 and t>0 , solve the one dimensional heat flow equation Thita u / Thita t = 4 thita^2 u / Thita x^2 satisfying the condition u(t,0) = u(t,5) = 0 , u(0,x) = x

The differential equation satisfied by a beam uniformly loaded ( W kg/ meter ) with one end fixed and the second end subjected to a tensile force-P is given by

El d^2 y/dx^2 = Py - 1/2 Wx^2

Where E is the modulus of elasticity and l is the moment of inertia show that the elastic curve for the beam with condition y = 0 and dy/dx = 0 at x = 0 is given by

y = W/Pn^2 (1-cosh nx ) + Wx^2 /2P where n^2 = ( P/EI)

Show that the complete integral of z = px +qy - 2p - 3q represent all possible planes through the points (2, 3, 0)

A mass weighing 39 5. kg. stretches a spring 1/4m. At t = 0 , the mass is released from a point 3/4m below the equilibrium position with an upward velocity of 5/4 m/sec.Determine the function x(t) that describes the subsequent free motion.

Find the surface which intersects the surfaces of the system z( x + y) = c(3z + 1) orthogonally and which passes through the circle x^2 + y^2 = 1 , z = 1

Find the integral surface of the partial differential equation

(x - y) y^2 p + (y - x) x^2 q = (x^2 + y^2) z

Through the curve xz = a^2 , y = 0

Apply the method of variations of parameters to solve the following differential equations:

1) x^2 y" + xy' - y = x^2 e^x

2) y" + a^2 y = cosec ax

3) solve the equation d^2 y/dx^2 - cotx dy/dx - sin^2 xy = cosx - cos^3 x by changing the independent variable

solve the following equation by changing the independent variable x^2 d^2 y/dx - dy/dx -4x^3 y = 8x^3 sin x^2 , x>0

1.1) y^-2 dy/dx + y^-1 = 2x


1.2) (y^2e^xy^2 +4x^3)dx + (2xye^xy^2 - 3y^2)dy = 0