Question

The differential equation representing the family of curves $y=x{e}^{cx}$ (c is a constant) is

• A. $\frac{dy}{dx}=\frac{y}{x}(1-log\frac{y}{x})$
• B. $\frac{dy}{dx}=\frac{y}{x}log\left(\frac{y}{x}\right)+1$
• C. $\frac{dy}{dx}=\frac{y}{x}(1+log\frac{y}{x})$
• D. $\frac{dy}{dx}+1=\frac{y}{x}log\frac{y}{x}$

Answer 1

Answer: (C)

$\frac{dy}{dx}=\frac{y}{x}(1+log\frac{y}{x})$

$y=x{e}^{cx}\Rightarrow \frac{y}{x}=e^{cx}$

$\log \frac{y}{x}=cx$

Now, $\frac{dy}{dx}=e^{cx}+x{e}^{cx}c=e^{log\frac{y}{x}}(1+log\frac{y}{x})$