Question

The differential equation of the curve for which the initial ordinate of any tangent is equal to the corresponding sub-normal is :

• A. Linear
• B. Homogenous
• C. Exact
• D. None of these

Answer 1

Answer: (A), (B)

The correct options are
A Linear
B Homogenous

We have, $2f\left(x\right)=f^{\prime }\left(x\right)\Rightarrow \frac{f^{\prime }\left(x\right)}{f\left(x\right)}=2.$
Integrating, we get $\log f\left(x\right)=2x+c_1.$
$\Rightarrow f\left(x\right)=e^{2x+c_1}=e^{c_1}.e^{2x}={ce}^{2x},$ where $c=e^{c_1}$
Putting $x=0,f\left(0\right)=3,$ we get $c=3.$
$\therefore f\left(x\right)={3e}^{2x}$

$\Rightarrow f\left(2\right)={3e}^4.$