Differential Equations Answers

Solve the following equation by changing the independent variable
(1+x^2)^2 y" +2x(1+x^2) y' + 4y=0

true/false.justify.
The normal form of the differential equation
y" - 4xy' + (4x^2 - 1)y =-3e^x^2 sin2x is
(d^2v/dx^2)+v = -3 sin2x, where v= y e^-x^2

The initial value problem
x y , )0(y 0
dx
dy 2 2
= + =
has a unique solution in some interval of the form − h < x < h .

Show that the wave equation a^2uxx = utt can be reduced to the form u xi×ita by the change of variable xi=x-at, ita = x+at.

Solve the following differential equations (i) [D^3-DD'^2-D^2+DD']z (ii) [D^4-D'^4-2D^2D'^2]z =0 (iii) [D^2-2DD'+D'^2]z =12xy.

Find the temperature in a bar of length l with both ends insulated and with initial temperature in the rod being sin πx/L.

Using the method of separation of variables, solve uxt = e^-1 cos x when u(x,0) =0, du/dt (0,t)= 0

Find the equation of the integral surface of the differential equation (x^2-yz)p+(y^2-zx)q= z^2-xy which passes through the line x =1, y =0.

Solve the differential equation (1) x^2p + y^2q = (x+y)z. (2) p^1/2 -q^1/2 +3x =0

Verify that the pfaffian differential equation yz dx+ (x^2y-zx) dy +(x^2z-xy) dz = 0 is integrable and hence find its integral.