MHT CET202525 Apr 2025Morning ShiftMathematicsDifferential EquationsActual
The differential equation x dy d x =2 y represents ....
Options
- Aa family of circles with radius c .
- Ba family of parabolas with vertex at the origin and axis along the positive Y-axis
- Ca family of parabolas with vertex the at origin and axis along the positive X-axis
- Da family of ellipses
Correct answer
B. a family of parabolas with vertex at the origin and axis along the positive Y-axis
Step-by-step solution
The differential equation x dy dx =2y separates as dy y = 2 x dx . Integration yields |y|=2 |x|+C₁ , which simplifies to |y|= |x^2|+C₁ . Writing C₁= |C| gives |y|= |Cx^2| , so exponentiating results in |y|=|Cx^2| . This represents the family of curves y=Cx^2 , where C is an arbitrary constant. These are parabolas with vertices at the origin and axes along the Y-axis. Comparing options: A: Family of circles (incorrect) B: Family of parabolas with vertex at origin along positive Y-axis (correct) C: Family with axis a