Question:
Which of the following is a solution of the differential equation \(\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}+4y=0\)?
Which of the following is a solution of the differential equation \(\dfrac{d^2y}{dx^2}+4\dfrac{dy}{dx}+4y=0\)?
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For repeated root \(m\) of the auxiliary equation, the solution contains both \(e^{mx}\) and \(xe^{mx}\).
Updated On: Jul 3, 2026
- \(e^{-3x}\)
- \(xe^{-x}\)
- \(xe^{-2x}\)
- \(x^2e^{-x}\)
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The Correct Option is C
Solution and Explanation
Concept:
For a linear differential equation with constant coefficients, we form the auxiliary equation by replacing \[ \frac{d}{dx} \] with \(m\). The given differential equation is \[ \frac{d^2y}{dx^2}+4\frac{dy}{dx}+4y=0 \]
Step 1: Form the auxiliary equation. Replace \[ \frac{d}{dx} \] by \(m\). Therefore, \[ m^2+4m+4=0 \]
Step 2: Factorize the auxiliary equation. \[ m^2+4m+4=(m+2)^2 \] So, \[ (m+2)^2=0 \] Hence, \[ m=-2,-2 \] This means \(m=-2\) is a repeated root.
Step 3: Write the general solution. If the auxiliary equation has repeated roots \(m,m\), then the solution is \[ y=(c_1+c_2x)e^{mx} \] Here, \[ m=-2 \] So, \[ y=(c_1+c_2x)e^{-2x} \]
Step 4: Identify one solution from the options. From the general solution, \[ y=e^{-2x} \] and \[ y=xe^{-2x} \] are solutions. Among the given options, the matching solution is \[ xe^{-2x} \]
Step 5: Final answer. \[ \boxed{xe^{-2x}} \]
For a linear differential equation with constant coefficients, we form the auxiliary equation by replacing \[ \frac{d}{dx} \] with \(m\). The given differential equation is \[ \frac{d^2y}{dx^2}+4\frac{dy}{dx}+4y=0 \]
Step 1: Form the auxiliary equation. Replace \[ \frac{d}{dx} \] by \(m\). Therefore, \[ m^2+4m+4=0 \]
Step 2: Factorize the auxiliary equation. \[ m^2+4m+4=(m+2)^2 \] So, \[ (m+2)^2=0 \] Hence, \[ m=-2,-2 \] This means \(m=-2\) is a repeated root.
Step 3: Write the general solution. If the auxiliary equation has repeated roots \(m,m\), then the solution is \[ y=(c_1+c_2x)e^{mx} \] Here, \[ m=-2 \] So, \[ y=(c_1+c_2x)e^{-2x} \]
Step 4: Identify one solution from the options. From the general solution, \[ y=e^{-2x} \] and \[ y=xe^{-2x} \] are solutions. Among the given options, the matching solution is \[ xe^{-2x} \]
Step 5: Final answer. \[ \boxed{xe^{-2x}} \]
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