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Solve the following.
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)[1.2]
Transform the above equation in the form $\frac{dy}{dx} = f\left(\frac{y}{x}\right)$.
(b)[1.2]
Use appropriate substitution to transform it into variable separable form.
(c)[1.2]
Write the differential equation in variable separable form.
(d)[1.2]
Prove that the solution of the differential equation is $\sec\left(\frac{y}{x}\right) = cxy$.
(e)[1.2]
Find the solution if $x = 1, y = 1$.
Answer
Answer (a)
Official answer key$\frac{dy}{dx} = \frac{\frac{y}{x}\left[\frac{y}{x}\tan\frac{y}{x} + 1\right]}{\frac{y}{x}\tan\frac{y}{x} - 1}$
Final answer: $\frac{\frac{y}{x}\left[\frac{y}{x}\tan\frac{y}{x} + 1\right]}{\frac{y}{x}\tan\frac{y}{x} - 1}$
Answer (b)
Official answer keyAppropriate Substitution: $y = vx$ or $\frac{y}{x} = v$
Final answer: $y = vx$
Answer (c)
Official answer key$\int\left(\tan v - \frac{1}{v}\right)dv = 2\int\frac{dx}{x} + c$
Final answer: $\int\left(\tan v - \frac{1}{v}\right)dv = 2\int\frac{dx}{x} + c$
Answer (d)
Official answer key$\sec\left(\frac{y}{x}\right) = cxy$ or $xy\cos\frac{y}{x} = A$
- $\int\left(\tan v - \frac{1}{v}\right)dv = 2\int\frac{dx}{x} + c$
- $\log\sec v - \log v = 2\log x + \log c$
- $\sec\left(\frac{y}{x}\right) = cxy$ or $xy\cos\frac{y}{x} = A$
Answer (e)
Official answer keyWhen $x = 1$, $y = 1$, $c = \sec(1) = 1.85$
Solution, $\sec\left(\frac{y}{x}\right) = 1.85xy$
When $x = 1$, $y = 1$, $A = \cos(1) = 0.54$
Solution, $xy\cos\left(\frac{y}{x}\right) = 0.54$
Final answer: $\sec\left(\frac{y}{x}\right) = 1.85xy$, i.e. $xy\cos\left(\frac{y}{x}\right) = 0.54$
From ISC 2025 Practice Mathematics, question 133.