Question

The differential equation obtained by eliminating arbitrary constants from $y = ae ^{ bx }$ is

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

Answer 1

Answer: (C)

The given equation is $y = ae ^{b x }$
$\Rightarrow \frac{ dy }{ dx }= abe ^{ bx } $
$\Rightarrow \frac{ d ^{2} y }{ dx ^{2}}= ab ^{2} e ^{ bx } \dots$(i)
$\Rightarrow a e^{b x} \frac{d^{2} y}{d x^{2}}=a^{2} b^{2} e^{b x} \dots$(ii)
$\Rightarrow y \frac{ d ^{2} y }{ dx ^{2}}=\left(\frac{ dy }{ dx }\right)^{2} [$ from eq. (ii) $]$
$\Rightarrow y \frac{d^{2} y}{d x^{2}}-\left(\frac{d y}{d x}\right)^{2}=0$