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Differential Equations — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDifferential Equations1 Mark
Question
The integrating factor of the differential equation $\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\text{x}^{2}.$
  1. $\text{e}^{-\text{x}}$
  2. $\text{e}^{-\text{y}}$
  3. $\frac{1}{\text{x}}$
  4. $\text{x}$
✓

Answer

  1. ​​​​​$\frac{1}{\text{x}}$

Solution:

We have,

$\text{x}\frac{\text{dy}}{\text{dx}}-\text{y}=2\text{x}^{2}$

$\Rightarrow \frac{\text{dy}}{\text{dx}}-\frac{1}{\text{x}}\text{y}=2\text{x}$

Comparing with $\frac{\text{dy}}{\text{dx}}+\text{Py}=\text{Q}$ we get,

$\text{P}=-\frac{1}{\text{x}}, \text{Q}=2\text{x}$

Now,

$\text{I.F}=\text{e}^{\int\frac{1}{\text{x}}\text{dy}}$

$=\text{e}^{\log|\text{x}|}$

$=\text{e}^{\log|\frac{1}{\text{x}}|}$

$=\frac{1}{\text{x}}$

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