Question

The family of curves passing through $(0,0)$ and satisfying the differential equation $\frac{y_2}{y_1}=k$, where $y_n=\frac{d^{n}y}{d{x}^n}$ is

• A. $y=k$
• B. $y=kx$
• C. $y=k(e^x+1)$
• D. $y=k(e^x-1)$

Answer 1

The correct option is D $y=k(e^x-1)$

Given that $y_n=\frac{d^{n}y}{d^{n}x}$,

Also a family of curves are passing through origin satisfying the differential equation $\frac{y_2}{y_1}=k$

$⟹\ddot{y}=k\dot{y}$,derivatives of $y$ are not zero

$⟹\int \ddot{y}=\int k\dot{y}$

$⟹\dot{y}=ky+c$ where '$c$' is the integration constant

$⟹\frac{dy}{dx}=ky+c$

$⟹\frac{dy}{ky+c}=dx$

integrating both sides gives,

$⟹\ln (ky+c)=x+p$,where '$p$' is the integration constant

As the curve passes through origin,

both the cnstant can be determined,

$⟹y=k(e^x-1)$ is equation of the family of curves where $k$ is a parameter.