AIEEE 2009 (Offline)Question 1
The differential equation which represents the family of curves $y=c_1e^{c_2x}$, where $c_1$ and $c_2$ are arbitrary constants, is
Differential Equation
Solution
Concept. Differentiate twice; $c_2$ appears in both derivatives and cancels in a ratio. Formulas used. (1) $y=c_1e^{c_2x}\Rightarrow y'=c_2y,\ y''=c_2^2y$Step 1 - Derivatives. By (1), $y'=c_2y$ and $y''=c_2^2y$. Step 2 - Eliminate. $yy''=c_2^2y^2=(c_2y)^2=(y')^2$. Answer: (C)