Question

The differential equation obtained by eliminating arbitrary constants from $y=a.e^{bx}$, is

• A. $y\frac{d^{2}y}{d{x}^2}+\frac{dy}{dx}=0$
• B. $y\frac{d^{2}y}{d{x}^2}-\frac{dy}{dx}=0$
• C. $y\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2=0$
• D. $y\frac{d^{2}y}{d{x}^2}+{\left(\frac{dy}{dx}\right)}^2=0$

Answer 1

The correct option is C $y\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2=0$

$y=a.e^{bx}$

two variables $\therefore$ second degree equation

diff. wrt x

$\frac{dy}{dx}a.e^{bx}.b.\Rightarrow \frac{dy}{dx}=y.b...\left(1\right)$

diff. again wrt x

$\frac{d^{2}y}{d{x}^2}=\frac{d}{dx}(y.b)=\frac{dy}{dx}.b$

$b=\frac{\left(\frac{d^{2}y}{d{x}^2}\right)}{\frac{dy}{dx}}\Rightarrow$ Placing in equation (1)

$\frac{dy}{dx}=y.\frac{\frac{d^{2}y}{d{x}^2}}{\left(dy/dx\right)}$

$\Rightarrow y.\frac{d^{2}y}{d{x}^2}-{\left(\frac{dy}{dx}\right)}^2=0$