Question

A differential equation associated to the primitive $y=a+b{e}^{5x}+c{e}^{-7x}$ is (where $y_n$ is $n^{th}$ derivative w.r.t. $x$)

• A. $y_3+2y_2-y_1=0$
• B. $4y_3+5y_2-20y_1=0$
• C. $y_3+2y_2-35y_1=0$
• D. $y_3+5y_2-20y_1=0$

Answer 1

The correct option is C $y_3+2y_2-35y_1=0$

Differentiating the given equation successively, we get
$y_1=5b{e}^{5x}-7c{e}^{-7x}\cdots \left(1\right)$
$y_2=25b{e}^{5x}+49c{e}^{-7x}\cdots \left(2\right)$
$y_3=125b{e}^{5x}-343c{e}^{-7x}\cdots \left(3\right)$
Multiplying equation $\left(1\right)$ by $7$ and then adding to equation $\left(2\right),$ we get
$y_2+7y_1=60b{e}^{5x}\cdots \left(4\right)$
Multiplying equation $\left(1\right)$ by $5$ and then subtracting it from equation $\left(2\right),$ we get
$y_2-5y_1=84c{e}^{-7x}\cdots \left(5\right)$
Putting the values of $b$ and $c,$ obtained from equation $\left(4\right)$ and $\left(5\right),$ respectively in equation $\left(3\right),$ we get
$y_3+2y_2-35y_1=0$