Question

The differential equation of the family of curve $y = A e^{3x} + B e^{5x}$, where A, B are arbitrary constants is :

• A. $\frac{d^2y}{dx^2}+8\frac{dy}{dx}=15y=0$
• B. $\frac{d^2y}{dx^2}-8\frac{dy}{dx}+15y=0$
• C. $\frac{d^2y}{dx^2}- \frac{dy}{dx}=y=0$
• D. None of these

Answer 1

Answer: (B)

$y = Ae^{3x }+ Be^{5x}$
$\Rightarrow \frac{dy}{dx} = 3Ae^{3x}+5Be^{5x}\quad\ldots\left(1\right)$
$\Rightarrow \frac{d^{3}y}{dx^{2}} = 9Ae^{3x}+25Be^{5x}\quad\ldots\left(2\right)$
$\therefore \frac{d^{2}y}{dx^{2}} - 8 \frac{dy}{dx}+15y$
$= 9\, Ae^{3x} + 25 \,Be^{5x} - 24Ae^{3x }- 40\, Be^{5x}$
$+ 15 \,Ae^{3x} + 15Be^{5x }= 0$