MHT CET202525 Apr 2025Morning ShiftMathematicsDifferential EquationsActual
y = e ^x( ~A x+ B x) is the solution of the differential equation
Options
- Ax^2 ~d ^2 y d x^2 + (1+ y ^2 )=0
- Bd ^2 y d x^2 - dy d x + y =0
- Cd ^2 y d x^2 -2 dy d x +2 y =0
- Dx ~d ^2 y d x^2 -2 dy d x +2 y =0
Correct answer
C. d ^2 y d x^2 -2 dy d x +2 y =0
Step-by-step solution
The differential equation for y = e^x(A x + B x) can be found by differentiating twice to eliminate the arbitrary constants A and B . Differentiating y with respect to x : dy dx = e^x(A x + B x) + e^x(-A x + B x) = y + e^x(-A x + B x) Differentiating again: d^2y dx^2 = dy dx + d dx [e^x(-A x + B x)] Applying the product rule: d^2y dx^2 = dy dx + e^x(-A x + B x) + e^x(-A x - B x) Substituting e^x(-A x + B x) = dy dx - y and e^x(-A x - B x) = -y : d^2y dx^2 = dy dx + ( dy dx - y) - y = 2 dy dx - 2y The resulting diff