Choose the correct answer from the given four option.Solution of … - Vidyadip

MCQ

Choose the correct answer from the given four option.Solution of $\frac{\text{d}\text{y}}{\text{d}\text{x}}-\text{y}=1,\text{ y}(0)=1$ is given by:

A
$\text{xy}=-\text{e}^\text{x}$
B
$\text{xy}=-\text{e}^{-\text{x}}$
C
$\text{xy}=-1$
✓
$\text{y}=2\text{e}^\text{x}-1$

✓

Answer

Correct option: D.

$\text{y}=2\text{e}^\text{x}-1$

Given is, $\frac{\text{d}\text{y}}{\text{d}\text{x}}-\text{y}=1$
$\Rightarrow\frac{\text{d}\text{y}}{\text{d}\text{x}}=\text{y}+1$
$\Rightarrow\frac{\text{d}\text{y}}{1+\text{y}}=\text{dx}$
On integrating both sides, we get
$\log(1+\text{y})=\text{x}+\text{C}\ ......(\text{i})$
When $x = 0$ and $y = 1,$ then
$\log2=0+\text{C}$
$\Rightarrow\text{C}=\log2$
The required solution is
$\log(1+\text{y})=\text{x}+\log2$
$\Rightarrow\log\Big(\frac{1+\text{y}}{2}\Big)=\text{x}$
$\Rightarrow\frac{1+\text{y}}{2}=\text{e}^\text{x}$
$\Rightarrow1+\text{y}=2\text{e}^\text{x}$
$\Rightarrow\text{y}=2\text{e}^\text{x}-1$

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