Question

The general solution of the differential equation $100\left(\frac{d^{2}y}{d{x}^2}\right)-20\left(\frac{dy}{dx}\right)+y=0$is

• A. $y=(c_1+c_{2}x)e^x$
• B. $y=(c_1+c_{2}x)e^{-x}$
• C. $y=(c_1+c_{2}x)e^{\frac{x}{10}}$
• D. $y=c_1{e}^x+c_2{e}^{-x}$

Answer 1

The correct option is C

$y=(c_1+c_{2}x)e^{\frac{x}{10}}$

Find the general solution of the differential equation.

Given,$100\left(\frac{d^{2}y}{d{x}^2}\right)-20\left(\frac{dy}{dx}\right)+y=0$

Put $D=\frac{d}{dx}$

$$\begin{gathered} \Rightarrow (100D^2-20D+1)y=0 \\ \Rightarrow 100D^2-20D+1=0 \\ \Rightarrow 100D^2-10D-10D+1=0 \\ \Rightarrow 10D(10D-1)-1(10D-1)=0 \\ \Rightarrow {(10D-1)}^2=0 \\ \Rightarrow D=\frac{1}{10},\frac{1}{10} \end{gathered}$$

Since, roots are real and equal.

$\therefore$General Solution is $y=(c_1+c_{2}x)e^{\frac{x}{10}}$

Hence, option (C) is the correct answer.