Q70 / 170 · Manipal (MET) 2016 · Mathematics
The differential equation of the family of curves $y=A e^{3 x}+B e^{5 x}$, where $A$ and $B$ are arbitrary constants, is
- A$\frac{d^2 y}{d x^2}+8 \frac{d y}{d x}+15 y=0$
- B$\frac{d^2 y}{d x^2}-\frac{d y}{d x}+y=0$
- Correct answerC$\frac{d^2 y}{d x^2}-8 \frac{d y}{d x}+15 y=0$
- DNone of the above
Solution
Given, $y=A e^{3 x}+B e^{5 x}$\ldots(i)
$\Rightarrow \quad \frac{d y}{d x}=3 A e^{3 x}+5 B e^{5 x}$\ldots(ii)
and $\quad \frac{d^2 y}{d x^2}=9 A e^{3 x}+25 B e^{5 x}$\ldots(iii)
From Eqs. (i), (ii) and (iii), we have
$\frac{d^2 y}{d x^2}-8 \frac{d y}{d x}+15 y=0$
which is the required differential equation.
Hence, the correct answer is option: 'C'.