Reference no: EM13912431
We have following set of differential equations
T0,1 V1 + 2T0 V(,0)0 = 0 (1)
T0 V(,1)0 + T1 V(,0)1 = 0 (2)
T0 V(,2)0 + T2 V(,0)2 = 0 (3)
T0 V(,3)0 +sin2θ T2 V(,0)3 = 0 (4)
T1,1 V1+ 2T1 V(,1)1 = 0 (5)
T1 V(,2)1+T2 V(,1)2 = 0 (6)
T1 V(,3)1+sin2θ T2 V(,1)3 = 0 (7)
V(,3)2+ sin2θ V(,2)3 = 0 (8)
T2,1 V1+ 2T2 V(,2)2 = 0 (9)
T2,1 V1+2T2 cot? θ V2 + 2T2 V(,3)3 = 0 (10)
NOTE :
1) Here comma denotes derivatives and indices 0,1,2,3 corresponds to variables t, r, , θ ,φ respectively & t, r, , θ , φ are independent variables.
2) T0 (r),T1 (r),T2 (r), are non-zero functions of r alone.
3) V0 , V1,V2, V3 are non-zero function of t, r, , θ ,φ . If we solve equations (1) to (10) , we have solution
V0= T2sinθ/T0.[A1sinφ - A2cosφ] + T2/T1 A3cosθ + A4 (11)
V1= T2sinθ/T1.[A'1sinφ - A'2cosφ] + T2/T1 A'3cosθ + A5 (12)
V2= cosθ.[A'1sinφ - A'2cosφ] + A3sinθ + c1sinφ - c2cosφ (13)
V3= cosecθ [A1cosφ - A2sinφ] + [c1cosφ + c2 sinφ]cotθ + c0 (14)
Where c0,c1,c2 are arbitrary constants and Aα= Aα (t,r),α = 1,2,3,4,5 Also prime on overhead of variables denotes differentiation with respect to " r " and dot on overhead of variables denotes differentiation with respect to " t "
These V0,V1,V2,V3 are satisfied subject to following differential constraints on Aα
2T1 A''i + T0,1A'i = 0, i =1, 2, 3 (15)
2T0 A'4 + T0,1 A5 = 0 (16)
2T2A'i + T0(T2/T0)' Ai = 0, (17)
T0 A'4 + T1 A5 = 0 (18)
{T1,1 T2/T1 + 2T1(T2/T1)'}A'i + 2T2Ai' = 0 (19)
T1,1A5 + 2T1A'5 = 0, (20)
T2,1A'i + 2T1Ai = 0, c0 = 0, (21)
T2,1A5 (22)
My questions :
1] what do you mean by these differential constraints.
2] how to obtain these solution using constrained equations (11) to (22).
It means give me all calculations to get equ. (11) to (22) in step-by-step and in detailed manner .
To find answer remember that there is only one condition that T0(r),T1 (r),T2 (r),T3 (r) and V0(t,r,,θ ,φ ), V1(t,r,,θ ,φ ), V2(t,r,,θ ,φ ), V3 (t,r,,θ ,φ ) are non-zero functions.