MHT CET202521 Apr 2025Morning ShiftMathematicsDifferential EquationsActual
The general solution of dy d x =2 x ye ^ x^2 is
Options
- Ay=e^ -e^ x^2 c, where c is the constant of integration
- By=e^ -x^2 c, where c is the constant of integration
- Cy = e ^ e ^ x^2 c , where c is the constant of integration
- Dy = e ^ x^2 c , where c is the constant of integration
Correct answer
C. y = e ^ e ^ x^2 c , where c is the constant of integration
Step-by-step solution
The separable differential equation is dy dx = 2xye^ x^2 . Separating variables yields dy y = 2xe^ x^2 ,dx . Integration gives |y| + C₁ = e^u ,du where u = x^2 and du = 2x ,dx , resulting in |y| + C₁ = e^ x^2 + C₂ . Combining constants and exponentiating: |y| = e^ e^ x^2 + C |y| = e^ e^ x^2 e^C Absorbing constants gives y = ce^ e^ x^2 , with c 0 including y = 0 as a solution when c = 0 . This matches option C . The answer is C .