The differential equation which represent the family of curves y = ae bx , where a and b are arbitrary constants.

Question

The differential equation which represent the family of curves y = ae bx , where a and b are arbitrary constants. (A) y '= y 2 (B) y'' = y y ' (C) y y '' = y' (D) y y''=(y')2

Tardigrade Admin · Accepted Answer

ln y = ln a + bx Differentiating w.r.t. x, we get: (1/y) y'=b Again differentiating w.r.t. x, we get (y''/y)-(1/y2)(y')2=0 ⇒ y y''=(y')2

Q. The differential equation which represent the family of curves $y = a e^{b x}$ , where $a$ and $b$ are arbitrary constants.

$y^{'} = y^{2}$

$y^{''} = y y^{'}$

$y y^{''} = y^{'}$

$y y^{''} = (y^{'})^{2}$

$ln y = ln a + b x$
Differentiating w.r.t. x, we get:
$\frac{1}{y} y^{'} = b$
Again differentiating w.r.t. $x$ , we get
$\frac{y ^{''}}{y} - \frac{1}{y ^{2}} (y^{'})^{2} = 0$
$\Rightarrow y y^{''} = (y^{'})^{2}$

Q. The differential equation which represent the family of curves y=aebx, where a and b are arbitrary constants.

y′=y2

y′′=yy′

yy′′=y′

yy′′=(y′)2