Question

The differential equation corresponding to the family of curves $y=e^x(ax+b)$ is

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

Answer 1

The correct option is D $\frac{d^{2}y}{d{x}^2}-2\frac{dy}{dx}+y=0$

We have $y=e^x(ax+b)$ ...(i)

Differentiating w.r.t. $x$, we get

$\frac{dy}{dx}=e^x(ax+b)+e^x.a$

or $\frac{dy}{dx}=y+a{e}^x$ ...........(ii)

Again, differentiating both the sides

$\frac{d^{2}y}{d{x}^2}=\frac{dy}{dx}+a{e}^x$ ..........(iii)

From Eq (iii) - Eq (ii)

$\frac{d^{2}y}{d{x}^2}-\frac{2dy}{dx}+y=0$

is the required differential equation.