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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS3 Marks
Question
For each of the differential equations in find the general solution:
$\frac{\text{dy}}{\text{dx}}=\frac{1-\text{cos} \ \text{x}}{1+\text{cos} \ \text{x}}$
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Answer

The given differential equation is
$\frac{\text{dy}}{\text{dx}}=\frac{1-\text{cosx}}{1+\text{cosx}}$
$\text{or}\ \ \ \ \text{dy} = \frac{1-\text{cosx}}{1+\text{cosx}}\text{dx}\ \ \ \text{or} \ \ \int\text{dy}=\int\frac{1-\text{cosx}}{1+\text{cosx}}\text{dx}$
$\therefore \ \ \ \int\text{dy} = \int \frac{2\ \text{sin}^2\frac{\text{x}}{2}}{2\ \text{cos}^2\frac{\text{x}}{2}}\text{dx} \ \text{or} \ \int1.\text{dy}=\int\text{tan}^2\frac{\text{x}}{2}\text{dx}$
$\text{or} \ \ \int 1.\text{dy}=\int\Big[\text{sec}^2\frac{\text{x}}{2}-1\Big] \text{dx}, \ \text{or}\ \text{y}=\frac{\text{tan}\frac{\text{x}}{2}}{\frac{1}{2}}-\text{x+c}$
$\text{or} \ \ \text{y}=2\text{tan}\frac{\text{x}}{2}-\text{x}+\text{c}$ which is the required solution.

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