If \( y=3e^{2x}+2e^{3x} \), then \( \displaystyle\frac{d^{2}y}{dx^{2}}-5\frac{dy}{dx} \) is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( -6y \)
Since \( y=3e^{2x}+2e^{3x} \),
\( \displaystyle y'=6e^{2x}+6e^{3x} \)
\( \displaystyle y''=12e^{2x}+18e^{3x} \)
Then
\( \displaystyle y''-5y'=12e^{2x}+18e^{3x}-30e^{2x}-30e^{3x} \)
\( \displaystyle =-18e^{2x}-12e^{3x} \)
\( \displaystyle =-6\left(3e^{2x}+2e^{3x}\right) \)
\( \displaystyle =-6y \)
Shortcut: 2 and 3 are the roots of \( m^{2}-5m+6=0 \), so \( y \) satisfies \( y''-5y'+6y=0 \).
Hence \( y''-5y'=-6y \) without differentiating.
NCERT (2026-27 edition):
Second Order Derivative
