Let y = y(x) be the solution of the differential equation ( x)^ 1/2 ,dy = ( ^3 x - ( x)^ 3/2 y) ,dx , 0 < x < 2 , y ( 4 ) = 6 2 5 . If y ( 3 ) = 4 5 , then ^4 equals _______.

Step-by-step solution

The given differential equation can be rewritten as: dy dx + ( x)y = ^3 x ( x)^ 1/2 This is a linear differential equation of the form dy dx + P(x)y = Q(x) . Integrating Factor (IF) = e^ x , dx = e^ ( x) = x Multiplying both sides by the IF and integrating, we get: y x = ^4 x ( x)^ 1/2 , dx + C y x = (1 + ^2 x) ^2 x ( x)^ 1/2 , dx + C Let x = t , then ^2 x , dx = dt . y x = 1 + t^2 t^ 1/2 , dt + C y x = (t^ -1/2 + t^ 3/2 ) , dt + C y x = 2t^ 1/2 + 2 5 t^ 5/2 + C y x = 2( x)^ 1/2 + 2 5 ( x)^ 5/2 + C Given y ( 4 ) =