Question

If y=\(\frac{x}{log_e|cx|}\) is the solution of the differential equation \(\frac{dy}{dx}=\frac{y}{x}+\phi(\frac{x}{y})\), then \(\phi (\frac{x}{y})\) is given by 

• A. \(\frac{y^2}{x^2}\)
• B. \(-\frac{y^2}{x^2}\)
• C. \(\frac{x^2}{y^2}\)
• D. \(-\frac{x^2}{y^2}\)

Answer 1

The Correct Option is A

We start with the differential equation:

dy/dx = xy + φ(y/x)

Let's make a substitution by setting y = vx:

Then, we find the derivative dx/dy:

dx/dy = v + x * (dv/dy)

Substituting this into the original equation, we have:

v + x * (dv/dy) = v + φ(v)

Now, the equation becomes:

x * (dx/dv) = φ(v) / φ'(v)

Integrating both sides, we get:

ln(x) = ln(φ(v)) + ln(k)

Solving this, we find:

kφ(v) = x

The correct answer is option (A): \(\frac{y^2}{x^2}\)