TS EAMCET · Maths · Differential Equations
Let \(p \in I R\), then the differential equation of the family of curves \(y=(\alpha+\beta x) e^{p x}\), where \(\alpha, \beta\) are arbitrary constants, is
- A \(y^{\prime \prime}+4 p y^{\prime}+p^2 y=0\)
- B \(y^{\prime \prime}-2 p y^{\prime}+p^2 y=0\)
- C \(y^{\prime \prime}+2 p y^{\prime}-p^2 y=0\)
- D \(y^{\prime \prime}+2 p y^{\prime}+p^2 y=0\)
Answer & Solution
Correct Answer
(B) \(y^{\prime \prime}-2 p y^{\prime}+p^2 y=0\)
Step-by-step Solution
Detailed explanation
Given, family of curve \(y=(\alpha+\beta x) e^{p x}\) On differentiating Eq. (i) w.r.t. \(x\), we get \[ \begin{aligned} \frac{d y}{d x} & =\beta \cdot e^{p x}+p e^{p x}(\alpha+\beta x) \\ \Rightarrow \quad y^{\prime} & =\beta \cdot e^{p x}+p y \end{aligned} \] [by Eq. (i)] On…
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