Question

Which of the following is the differential equation whose auxiliary roots are \(0,-1,-1\)?

• A. \(\dfrac{d^3y}{dx^3}+\dfrac{d^2y}{dx^2}-\dfrac{dy}{dx}+y=e^x\)
• B. \(\dfrac{d^3y}{dx^3}+2\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}+y=e^{-x}\)
• C. \(\dfrac{d^3y}{dx^3}+2\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=e^{-x}\)
• D. \(\dfrac{d^3y}{dx^3}-2\dfrac{d^2y}{dx^2}+\dfrac{dy}{dx}=e^{-x}\)

Hint:

Auxiliary roots \(0,-1,-1\) give the polynomial \(m(m+1)^2=m^3+2m^2+m\).

Answer 1

The Correct Option is C

Concept:
If the auxiliary roots of a linear differential equation are known, then the auxiliary polynomial is formed by multiplying the factors corresponding to those roots. If roots are \[ m=\alpha,\beta,\gamma \] then the auxiliary equation is \[ (m-\alpha)(m-\beta)(m-\gamma)=0 \]

Step 1: Write the given roots. The auxiliary roots are \[ 0,-1,-1 \] So the factors are \[ m-0=m \] and \[ m-(-1)=m+1 \] Since \(-1\) is repeated, we get \[ m(m+1)^2=0 \]

Step 2: Expand the auxiliary equation. \[ m(m+1)^2=0 \] \[ m(m^2+2m+1)=0 \] \[ m^3+2m^2+m=0 \]

Step 3: Convert auxiliary equation to differential equation. Replace \(m\) by \(D\), where \[ D=\frac{d}{dx} \] So, \[ D^3+2D^2+D \] Thus the differential operator is \[ \frac{d^3}{dx^3}+2\frac{d^2}{dx^2}+\frac{d}{dx} \]

Step 4: Match with the options. The matching differential equation is \[ \frac{d^3y}{dx^3}+2\frac{d^2y}{dx^2}+\frac{dy}{dx}=e^{-x} \]

Step 5: Final answer. \[ \boxed{\frac{d^3y}{dx^3}+2\frac{d^2y}{dx^2}+\frac{dy}{dx}=e^{-x}} \]