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NDA 2022
Exam
NDA
Year
2022
Subject
Mathematics
Chapter
Differential Equations

NDA 2022 · Mathematics · Differential Equations

What is the solution of the differential equation (d y-d x)+ x(d...

What is the solution of the differential equation (dy−dx)+cos⁡x(dy+dx)=0?(d y-d x)+\cos x(d y+d x)=0 ?

Options

  1. Ay=tan⁡(x2)−x+cy=\tan \left(\frac{x}{2}\right)-x+c
  2. By=12tan⁡(x2)−x+cy=\frac{1}{2} \tan \left(\frac{x}{2}\right)-x+c
  3. Cy=2tan⁡(x2)−x+cy=2 \tan \left(\frac{x}{2}\right)-x+c
  4. Dy=tan⁡(x2)−2x+cy=\tan \left(\frac{x}{2}\right)-2 x+c

Correct answer

C. y=2tan⁡(x2)−x+cy=2 \tan \left(\frac{x}{2}\right)-x+c

Step-by-step solution

(dy−dx)+cos⁡x(dy+dx)=0⇒(1+cos⁡x)dy=(1−cos⁡x)dx⇒dydx=1−cos⁡x1+cos⁡x=tan⁡2x2=sec⁡2x2−1∫dy=∫(sec⁡2x2−1)dx⇒y=2tan⁡x2−x+C\begin{aligned} & (d y-d x)+\cos x(d y+d x)=0 \\ & \Rightarrow(1+\cos x) d y=(1-\cos x) d x \\ & \Rightarrow \frac{d y}{d x}=\frac{1-\cos x}{1+\cos x}=\tan ^2 \frac{x}{2}=\sec ^2 \frac{x}{2}-1 \\ & \int d y=\int\left(\sec ^2 \frac{x}{2}-1\right) d x \\ & \Rightarrow y=2 \tan \frac{x}{2}-x+C\end{aligned}
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