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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS1 Mark
Question
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:
  1. $\text{xy}=-\text{e}^\text{x}$
  2. $\text{xy}=-\text{e}^{-\text{x}}$
  3. $\text{xy}=-1$
  4. $\text{y}=2\text{e}^\text{x}-1$
✓

Answer

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

Solution:

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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