system of 2 linear differential equations with variable coefficients f ″ ( x ) + a f ′ ( x ) + ( 1 + x ) g ′ ( x ) + b g ( x ) = 0 g ″ ( x ) + a g ′ ( x ) + ( 1 − x ) f ′ ( x ) − b f ( x ) = 0where a , b are some positive constants.

Question

system of 2 linear differential equations with variable coefficients      f    ″    (  x  )  +  a      f    ′    (  x  )  +  (  1  +  x  )      g    ′    (  x  )  +  b  g  (  x  )  =  0        g    ″    (  x  )  +  a      g    ′    (  x  )  +  (  1  −  x  )      f    ′    (  x  )  −  b  f  (  x  )  =  0where   a  ,  b are some positive constants.

pradassas66b2d · Accepted Answer

Let       f    ′    (  x  )  =  h  (  x  ) and       g    ′    (  x  )  =  k  (  x  ). The system of second order ODE can be reduced to an homogeneous system of first order with variable coefficients       z    ′    (  x  )  +  A  (  x  )  z  (  x  )  =  0:      (                                        f            ′                                                            g            ′                                                            h            ′                                                            k            ′                                )    +                      (                                            0                                      0                                      −              1                                      0                                                          0                                      0                                      0                                      −              1                                                          0                                      0                                      a                                      1              +              x                                                          −              b                                      0                                      1              −              x                                      a                                      )            ⏟              A      (      x      )                               (                                            f                                                          g                                                          h                                                          k                                      )            ⏟              z      (      x      )        =      (                            0                                      0                                      0                                      0                      )  Thus you can solve it by Fundamental System of Solutions and Fundamental Matrix Method (Wronskian and Liouville&#039;s Formula) or by Method of Variation of Constants (Lagrange Method)

system of 2 linear differential equations with variable coefficients f ″ <

Answered question

Answer & Explanation

New Questions in Algebra I