Question

The solution of the differential equation is
$\frac{d^{2}y}{d{x}^2}=\frac{dy}{dx}$
Where $C_{1}and{C}_2$ are arbitrary constants.

• A. $y=C_1+C_2{e}^{-x}$
• B. $y=C_1{e}^x$
• C. $y=C_1+C_2{e}^x$
• D. $y=C_2{e}^{-x}$

Answer 1

The correct option is C $y=C_1+C_2{e}^x$

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

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

Let $\frac{d}{dx}=m(\because y\ne 0)$

$\Rightarrow (m^2-m)y=0$

$\Rightarrow m(m-1)=0$

$\Rightarrow m_1=0and{m}_2=1$

$y=C_1{e}^{m_{1}x}+C_2{e}^{m_{2}x}$

$=C_1{e}^0+C_2{e}^x$

$y=C_1+C_2{e}^x$