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DIFFERENTIAL EQUATIONS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsDIFFERENTIAL EQUATIONS5 Marks
Question
Find the general solution of the differential equation $\text{x}\cos\Big(\frac{\text{y}}{\text{x}}\Big)\frac{\text{dy}}{\text{dx}}=\text{y}\cos\Big(\frac{\text{y}}{\text{x}}\Big)+\text{x}.$
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Answer

Given differential equation can be written as $\frac{\text{dy}}{\text{dx}} = \frac{\text{y}}{\text{x}} + \frac{1}{\cos\big(\frac{\text{y}}{\text{x}}\big)}$ $\text{put y = vx} \Rightarrow \frac{\text{dy}}{\text{dx}} = \text{v + x} \frac{\text{dv}}{\text{dx}}$ $\therefore \text{v + x} \frac{\text{dv}}{\text{dx}} = \text{v} + \frac{1}{\cos{\text{v}}}$ $\Rightarrow \int{\cos\text{v}}\ {\text{dv}} = \int \frac{\text{dx}}{\text{x}}$ $\Rightarrow \sin\text{v}=\log |\text{x}| +\text{c}$$\Rightarrow\ \sin\big(\frac{\text{y}}{\text{x}}\big)=\log|\text{x}|+\text{c}$

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