Question:
The differential equation of the family \( y = a e^{x} + b x e^{x} + c x^{2} e^{x} \) is
The differential equation of the family \( y = a e^{x} + b x e^{x} + c x^{2} e^{x} \) is
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A repeated root 'r' of multiplicity 'k' in a linear ODE leads to solutions $e^{rx}, xe^{rx}, ..., x^{k-1}e^{rx}$.
Updated On: Apr 10, 2026
- $y''' + 3y'' + 3y' + y = 0$
- $y''' + 3y'' - 3y' - y = 0$
- $y''' - 3y'' - 3y' + y = 0$
- $y''' - 3y'' + 3y' - y = 0$
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The Correct Option is D
Solution and Explanation
Step 1: Formation of Characteristic Equation
The solutions $e^x, xe^x, x^2e^x$ correspond to a root $m=1$ repeated three times. The auxiliary equation is $(m-1)^{3} = 0$.
Step 2: Expansion
$m^{3} - 3m^{2} + 3m - 1 = 0$.
Step 3: Conversion to Differential Equation
Replace powers of $m$ with corresponding derivatives: $y''' - 3y'' + 3y' - y = 0$.
Final Answer: (d)
The solutions $e^x, xe^x, x^2e^x$ correspond to a root $m=1$ repeated three times. The auxiliary equation is $(m-1)^{3} = 0$.
Step 2: Expansion
$m^{3} - 3m^{2} + 3m - 1 = 0$.
Step 3: Conversion to Differential Equation
Replace powers of $m$ with corresponding derivatives: $y''' - 3y'' + 3y' - y = 0$.
Final Answer: (d)
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