The solution of the differential equation dy dx -Ky=0, y(0)=1 app… - Vidyadip

MCQ

The solution of the differential equation $\frac{\text{dy}}{\text{dx}}-\text{Ky}=0, \text{y}(0)=1$ approaches to zero when $\text{x}\rightarrow\propto$ if,

A
$K = 0$
B
$K > 0$
✓
$K < 0$
D
None of these.

✓

Answer

Correct option: C.

$K < 0$

We have,
$\Rightarrow \frac{\text{dy}}{\text{dx}}-\text{Ky}=0$
$\Rightarrow \frac{\text{dy}}{\text{dx}}=\text{Ky}$
$\Rightarrow \frac{1}{\text{y}}\text{dy}=\text{K}\ \text{dx}$
Integrating both sides, we get
$ \int\frac{1}{\text{y}}\text{dy}=\text{K}\int\text{dx}$
$\Rightarrow \log|\text{y}|=\text{Kx}+\text{C}\ ...(\text{i})$
Now,
$\text{y}(0)=1$
$\text{C}=0$
Putting $C = 0$ in $(i),$
$\log|\text{y}|=\text{Kx}$
$\Rightarrow \text{e}^{\text{Kx}}=\text{y}$
According to the quation,
$\text{e}^{\text{K}\propto}=0$

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