MHT CET202615 April 2026Evening ShiftMathematicsDifferential EquationsActual
The solution of d y d x = (x + y) + (x + y) is
Options
- A[1 + ( x+y 2 ) ] = y + c , where c is the constant of integration
- B[1 - ( x+y 2 ) ] = y + c , where c is the constant of integration
- C[1 + ( x+y 2 ) ] = x + c , where c is the constant of integration
- D[1 - ( x+y 2 ) ] = x + c , where c is the constant of integration
Correct answer
C. [1 + ( x+y 2 ) ] = x + c , where c is the constant of integration
Step-by-step solution
Let x + y = v Differentiating with respect to x , we get 1 + d y d x = d v d x d y d x = d v d x - 1 Substituting in the given differential equation: d v d x - 1 = v + v d v d x = 1 + v + v d v 1 + v + v = d x Using the half-angle formulas v = 2 (v/2) 1+ ^2(v/2) and v = 1- ^2(v/2) 1+ ^2(v/2) : d v 1 + 2 (v/2) 1+ ^2(v/2) + 1- ^2(v/2) 1+ ^2(v/2) = d x (1+ ^2(v/2)) d v 1+ ^2(v/2) + 2 (v/2) + 1- ^2(v/2) = d x ^2(v/2) d v 2(1+ (v/2)) = d x Integrating both sides: ^2(v/2) 2(1+ (v/2)) d v = d x Let 1 + (v/2) = t , then 1