If y = y ( x ) is the solution curve of the differential equation x 2 - 4 dy - y 2 - 3 y dx = 0 , x > 2 , y ( 4 ) = 3 2 and the slope of the curve is never zero, then the value of y ( 10 ) equals :

Step-by-step solution

Given, x 2 - 4 dy - y 2 - 3 y dx = 0 ⇒ ∫ dy y 2 - 3 y = ∫ dx x 2 - 4 ⇒ 1 3 ∫ y - ( y - 3 ) y ( y - 3 ) dy = ∫ dx x 2 - 4 ⇒ 1 3 ∫ 1 y - 3 - 1 y dy = ∫ dx x 2 - 2 2 ⇒ 1 3 log | y - 3 | - log | y | = 1 4 log x - 2 x + 2 + C ⇒ 1 3 log y - 3 y = 1 4 log x - 2 x + 2 + C At x = 4 , y = 3 2 ⇒ 1 3 log 3 2 - 3 3 2 = 1 4 log 4 - 2 4 + 2 + C ⇒ 0 = 1 4 log 1 3 + C ⇒ 0 = - log 3 1 4 + C ⇒ C = 1 4 log 3 ⇒ 1 3 log y - 3 y = 1 4 log x - 2 x + 2 + 1 4 log ( 3 ) Putting, x = 10 ⇒ 1 3 log y - 3 y = 1 4 log 2 3 + 1 4 log ( 3 ) ⇒ 1 3 lo