Question

The solution of the differential equation \[ \frac{d^3y}{dx^3}+3\frac{d^2y}{dx^2}+2\frac{dy}{dx}=0 \] is

• A. \(y=a+be^{-x}+ce^{-2x}\)
• B. \(y=a+b e^{x}+ce^{2x}\)
• C. \(y=ae^{-x}+be^{-2x}+ce^x\)
• D. \(y=a+be^{-2x}+ce^{-3x}\)

Hint:

For linear differential equations with constant coefficients, form the auxiliary equation and find roots.

Answer 1

The Correct Option is A

Step 1: Write auxiliary equation: \[ m^3+3m^2+2m=0 \]

Step 2: Factorize: \[ m(m^2+3m+2)=0 \] \[ m(m+1)(m+2)=0 \]

Step 3: Roots are: \[ m=0,\quad m=-1,\quad m=-2 \]

Step 4: Therefore, complementary function: \[ y=a e^{0x}+be^{-x}+ce^{-2x} \] \[ y=a+be^{-x}+ce^{-2x} \] \[ \boxed{y=a+be^{-x}+ce^{-2x}} \]