MHT CET202522 Apr 2025Evening ShiftMathematicsDifferential EquationsActual
The general solution of the differential equation. dy d x + ( x+ y 2 )= ( x- y 2 ) is
Options
- A( y 2 )=c-2 x 2 , where c is the constant of integration
- B( y 4 )= c -2 ( x 2 ) , where c is the constant of integration
- C[ ( y 2 + 4 ) ]= c -2 x , where c is the constant of integration
- D[ ( y 4 + 4 ) ]= c -2 x 2 , where c is the constant of integration
Correct answer
B. ( y 4 )= c -2 ( x 2 ) , where c is the constant of integration
Step-by-step solution
The differential equation is d y d x + ( x+y 2 ) = ( x-y 2 ) . Rearranging terms gives d y d x = ( x-y 2 ) - ( x+y 2 ) . Applying the sum-to-product identity A - B = 2 ( A+B 2 ) ( A-B 2 ) with A = x-y 2 and B = x+y 2 yields d y d x = -2 ( x 2 ) ( y 2 ) . Separating variables results in d y ( y 2 ) = -2 ( x 2 ) d x . Integrating both sides: ( y 2 ) d y = -2 ( x 2 ) d x The left side integrates to 2 | ( y 4 ) | and the right side to -4 ( x 2 ) , giving 2 | ( y 4 ) | = -4 ( x 2 ) + C' . Simplifying yields | ( y 4 ) |