Differential Equations Answers

Solve the following DEs
(i) [(dy/dx)-1]^2 [(d^2y/dx^2)+1]^2 y= sin^2 (x/2) +e^x +x
(ii) 2x^2y (d^2y/dx^2) + 4y^2 = x^2 (dy/dx)^2 + 2xy (dy/dx)

Solve the following IVP
(d^2y)/(dx^2) + (dy)/(dx) - 2y = -6 sin2x -18 cos2x
y(0)=2, y'(0)=2

Find the integrating factor of the differential equation
(6xy-3y^2+2y) dx + 2(x-y) dy =0
and hence solve it

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 electrostatic potential V(x,y) in a rectangular region is governed by the two-
dimensional Laplace equation:
d^2V (x,y)/dx^2 + d^2v (x,y)/dy^2 =0
Determine V(x,y) in a rectangular region 0 ≤ x ≤ 20 and 0 ≤ y ≤ 40 for the following
boundary conditions:
V ,0( y) =V 20( , y) =V(x )0, = 0
and
V (x 40, ) =110V

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.