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Q.
The differential equation which represent the family of curves $y = ae ^{ bx }$, where $a$ and $b$ are arbitrary constants.
Differential Equations
Solution:
$\ln \,y =\ln \,a + bx$
Differentiating w.r.t. x, we get:
$\frac{1}{y} y'=b$
Again differentiating w.r.t. $x$, we get
$\frac{y''}{y}-\frac{1}{y^{2}}\left(y'\right)^{2}=0$
$ \Rightarrow y y''=\left(y'\right)^{2}$