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If F(-a^2)=0, then prove that:


i. (1/F(D^2))sinax=x(1/d/dD[F(D^2)])sinax

ii.

(1/F(D^2))cosax=x(1/d/dD[F(D^2)])cosax
Differential Operators and Fundamental Laws of Operation


Prove that AB=A(By)=B(Ay)


Let A=xD+2 and B=xD-1

Solve y"' -5y"+ 4y'=8ex + 4x.



Find y as a function of t

 if 16y′′ + 136y′ +289y = 0,

y(0) = 7 , y′(0) = 3.



  1. y(3x^3-x+y)dx+x^2(1-x^2)dy=0
  2. y(2x+y^2)dx+x(y^2-x)dy=0
Find the general integrals of the PDE
(Y+1)p+(x+1)q=z
Find complete integral of the equation
p+q=pq
Find the characteristics of the equation pq=xy
And determine the integral surface passes through the curve z=x , y=0
Solve (D2
-3D+2)y = 2cos (2x+3)+2 ex
.


Find the general solution of the differential equation

(D^2+1)y=x^3+e^xcosx