Differential Equations Answers

Solve the following system of ODE

x'+x-y = e^2t

Let G be a finite group and H be a subgroup of G with odd number of
elements such that (G:H) = 2. Show that the product of all elements of G (taken
in any order) does not belong to H.

Solve
(p+q)(px+qy)=1,using charpit's method

[(D-1)^2.(D^2+1)^2]y=sin^2x/2+e^x

The solution of the differential equation
dy/dx=y with y(0)=0
exists,but is not unique.
true or false with complete explanation

Find the integral field of the vector field given by X(p)=(p,2(y-b),-2(x-a) through the point (a+r,b) and f(x,y)=(x-a)^2+(y-b)^2 of a circle C=f^-1(r^2)

Solve the following differential equation:

x^2 d^2y/dx^2 - x dy/dx + 4y = cos(logx) + x sin(logx)

P^2z+q^2=1