Differential Equations Answers

Using the method of undetermined coefficients, write the trial solution of the equation
d^2y/dx^2 + 2dy/dx +5y= xe^-x cos2x

and hence solve it.

Solve the following equation :
(1) (D^2 - 2DD' + [D']^2 )z = tan(x - y)
(2) z(p - q) = z^2 + (x + y^2)

Find the value of n for which the equation (n - 1)^2 uxx - y^2n uyy = ny^2n-1 uy , is
(1) parabolic
(2) hyperbolic

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example.
(1) Equation cos(x+y) p+sin(x+y) q=z^2+z is a quasi -linear equation.
(2) The solution of PDE dz/dx+dz/dy=z^2 is z=-[y+f(x-y) ].
(3) dz/dx.dz/dy-(dz/dy) ^2=0is a non -linear PDE.

State whether the following statements are true or false. Justify yourself with the help of a short proof or a counter example.
(1) Equation cos(x+y) p+sin(x+y) q=z^2+z is a quasi -linear equation.
(2) The solution of PDE dz/dx+dz/dy=z^2 is z=-[y+f(x-y) ].
(3) dz/dx.dz/dy-(dz/dy) ^2=0is a non -linear PDE.

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.