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Consider the following differential equation and answer the questions: Transform the above equation…
Consider the following differential equation and answer the questions:
$\left[x\cos\left(\frac{y}{x}\right) + y\sin\left(\frac{y}{x}\right)\right] y\,dx - \left[y\sin\left(\frac{y}{x}\right) - x\cos\left(\frac{y}{x}\right)\right] x\,dy = 0$
(a)
Transform the above equation in the form $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$.
(b)
Use appropriate substitution to transform it into variable separable form.
(c)
Write the differential equation in variable separable form.
(d)
Prove that the solution of the differential equation is $\sec\left(\frac{y}{x}\right) = cxy$.
(e)
Find the solution if $x = 1, y = 1$.
Answer
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From ISC 2025 Practice Mathematics, question 133.