MHT CET202611 April 2026Evening ShiftMathematicsDifferential EquationsActual
The solution of the differential equation x dy dx + y = x^3 y^6 , is...
Options
- Ay^5 = 5 2 x⁻³ + cx^5
- By^5 = 5 2 x^3 + cx^5
- C1 y^5 = 5 2 x⁻³ + cx^5
- D1 y^5 = 5 2 x^3 + cx^5
Correct answer
D. 1 y^5 = 5 2 x^3 + cx^5
Step-by-step solution
The given differential equation is: x dy dx + y = x^3 y^6 Dividing the entire equation by x y^6 : 1 y^6 dy dx + 1 x y^5 = x^2 Let v = 1 y^5 = y⁻⁵ . Differentiating with respect to x : dv dx = -5 y⁻⁶ dy dx 1 y^6 dy dx = - 1 5 dv dx Substituting this into the differential equation: - 1 5 dv dx + 1 x v = x^2 dv dx - 5 x v = -5 x^2 This is a linear differential equation of the form dv dx + P v = Q , where P = - 5 x and Q = -5 x^2 . The Integrating Factor (IF) is: IF = e^ - 5 x dx = e^ -5 x = x⁻⁵ The solution is given b