Coupled system of partial differential equations - give me

Question

Consider coupled system of partial differential equations which are first order and linear is given below

v_tX⁰ + v_rX¹ + X_t⁰ = ψ                          (1)

e^2λX_t¹ - e^2vX_r⁰ = 0                             (2)

Y²X_t² - e^2vX_θ⁰ = 0                             (3)

Y²sin²θX_t³ - e^2vX_Φ⁰ = 0                      (4)

λ_tX⁰ + λ_rX¹ + X_r¹ = ψ                          (5)

Y²X_r² + e^2λ X_θ¹ = 0                            (6)

Y²sin²θX_r³ + e^2λX_Φ¹ = 0                     (7)

Y_t/Y.X⁰ + Y_r/YX¹ +X_θ² = ψ                  (8)  

sin²θX_θ³ + X_Φ² = 0                             (9)

Y_t/Y.X⁰ + Y_r/YX¹ + cotθX² + X_Φ³ = ψ    (10)

Here subscripts denote partial differentiation'

X⁰, X¹, X², X³, ψ are non-zero function of r, θ, Φ, t.

In this process the equations tiecoupie, anti nuniber of integrability condition& are generated. Which gives General Solution as

X⁰ = Y²e^-2v sinθ(C_t sinΦ - D_t cosΦ) - Y²e^-2vI_t cosθ + J                                         (11)

X^I =  -Y²e^-2λ sinθ(C_r sinΦ - D_r cosΦ) - Y²e^-2λ I_r cosθ + K                                      (12)

X² = cosθ [C sinΦ- DcosΦ] + cosθ(a₁ sinΦ - a₂ cosΦ - a₃ sinΦ + a₄ cosΦ + Isinθ       (13)

X³ = cscθ [C cosΦ- DsinΦ] + cscθ(a₁ cosΦ + a₂ sinΦ) - cotθ(a₃ cosΦ + a₄ sinΦ) + a₅     (14)

ψ = YsinθsinΦ [Ye^-2vC_tt + (2Y_t - Y_vt)e^-2vC_t - Ye^-2λv_rC_r]

      - YsinθcosΦ [Ye^-2vD_tt + (2Y_t - Y_vt)e^-2vD_t - Ye^-2λv_rD_r]

      - Ycosθ [Ye^-2vI_tt + (2Y_t - Y_vt)e^-2vI_t - Ye^-2λv_rI_r]

      + J_t + v_tJ + v_rK                                                                                          (15)

where A, C, D, I, J, K, are functions t and r and a1 -a5 are constants. This general solution is subject to the following twelve consistency conditions

YC_tr + (Y_r - Yv_r)C_t + (Y_t - Yλ_t)C_r = 0                                                                  (16)

YD_tr + (Y_r - Yv_r)D_t + (Y_t - Yλ_t)D_r = 0                                                                  (17)

YI_tr + (Y_r - Yv_r)I_t + (Y_t - Yλ_t)I_r = 0                                                                    (18)

Ye^-2vC_tt + Ye^-2λC_{rr +} (2Y_t - Yλ_t - Yv_t)e^-2vC_r + (2Y_r - Yλ_r - Yv_r)e^-2λC_r = 0                (19)

Ye^-2vD_tt + Ye^-2λD_{rr +} (2Y_t - Yλ_t - Yv_t)e^-2vD_t + (2Y_r - Yλ_r - Yv_r)e^-2λD_r = 0                (20)

Ye^-2vI_tt + Ye^-2λI_{rr +} (2Y_t - Yλ_t - Yv_t)e^-2vI_t + (2Y_r - Yλ_r - Yv_r)e^-2λI_r = 0                   (21)

 

Y²e^-2vC_tt + Y(Y_t - Yv_t)e^-2vC_t + Y(Y_r - Yv_r)e^-2λC_r + C + a₁ = 0                               (22)

Y²e^-2vD_tt + Y(Y_t - Yv_t)e^-2vD_t + Y(Y_r - Yv_r)e^-2λD_r + D + a₂ = 0                               (23)

Y²e^-2vI_tt + Y(Y_t - Yv_t)e^-2vI_t + Y(Y_r - Yv_r)e^-2λI_r + I = 0                                          (24)

e^-2λK_t - e^2vJ_r = 0                                                                                             (25)

-J_t + (Y_t/Y - vt)J + (Y_r/Y - v_r) K = 0                                                                    (26)

-J_t + K_r + (λ_t - v_t)J + (λ_r - v_r) K = 0                                                                    (27)

Question

Give me detailed calculations of equations from 11 to 27 in step by step manner. It means how to obtain equ.11 to 27 in detailed manner.