Question

The differential equation satisfying all the curves $y=a{e}^{2x}+b{e}^{-3x}$ ,where a and b are arbitrary constants, is

• A. $6y=y_1-y_2$
• B. $y=y_1+y_2$
• C. $6y=y_1+y_2$
• D. None of these

Answer 1

The correct option is C $6y=y_1+y_2$

$y=a{e}^{2x}+b{e}^{-3x}$ …….$\left(1\right)$

Differentiating,

$y_1=2a{e}^{2x}-3b{e}^{-3x}$ ………..$\left(2\right)$

$\left(2\right)-2\times \left(1\right)$

$y_1-2y=-5b{e}^{-3x}$ ………..$\left(3\right)$

differentiating $\left(3\right)$

$y_2-2y_1=15b{e}^{-3x}$ ………$\left(4\right)$

$\left(4\right)+3\times \left(3\right)$

$y_2-2y_1+3y_1-6y=15b{e}^{-3x}-15b{e}^{-3x}$

$y_2+y_1=6y$.