TS EAMCET · Maths · Differential Equations
The differential equation of the family \(y=a e^x+b x e^x+c x^2 e^x\) of curves, where \(a, b, c\) are arbitrary constants, is
- A \(y^{\prime \prime \prime}+3 y^{\prime \prime}+3 y^{\prime}+y=0\)
- B \(y^{\prime \prime \prime}+3 y^{\prime \prime}-3 y^{\prime}-y=0\)
- C \(y^{\prime \prime \prime}-3 y^{\prime \prime}-3 y^{\prime}+y=0\)
- D \(y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0\)
Answer & Solution
Correct Answer
(D) \(y^{\prime \prime \prime}-3 y^{\prime \prime}+3 y^{\prime}-y=0\)
Step-by-step Solution
Detailed explanation
Given, \(y=a e^x+b x e^x+c x^2 e^x\) ...(i) On differentiating w.r.t. \(x\), we get \(\begin{aligned} y^{\prime} & =a e^x+b\left(x e^x+e^x\right)+c\left(x^2 e^x+2 x e^x\right) \\ \Rightarrow \quad y^{\prime} & =a e^x+b x e^x+c x^2 e^x+b e^x+2 c x e^x \end{aligned}\)…
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