Question

The number of solutions of the equation \(xydx— (x2-y2)dy=0\) with \( y(2)=3\) is :

• A. None
• B. One
• C. Two
• D. Infinite

Answer 1

The Correct Option is B

To find the number of solutions for the differential equation \(xydx - (x^2 - y^2)dy = 0\) with the initial condition \(y(2) = 3\), we proceed by checking the type of differential equation and solving it. First, rewrite the equation:

\(xydx = (x^2 - y^2)dy\)

Rearranging terms, we have:

\(\frac{xydx}{dy} = x^2 - y^2\)

This can be rewritten as:

\(\frac{dy}{dx} = \frac{xy}{x^2 - y^2}\)

The equation is separable, as we can express it as:

\(\int \frac{x}{x^2 - y^2}dx = \int \frac{y}{x^2 - y^2}dy\)

The integrals can be computed by appropriate substitutions. For simplicity, assume \(x = Cy\), substituting and simplifying leads to a general solution. Then, apply the given condition \(y(2) = 3\) to find the constant:

\(\int \frac{dy}{y} = \int \frac{dx}{x}\)

This yields:

\(\ln|y| = \ln|x| + C\)

With the initial condition \(y(2) = 3\), we find:

\(\ln|3| = \ln|2| + C\)

\(C = \ln\left(\frac{3}{2}\right)\)

Substituting back gives the unique exact solution:

\(|y| = \left|\frac{3}{2}x\right|\)

This calculation shows that there is only one solution, as any solution would have to satisfy the given initial condition.