Question

The solution of the differential equation \( (x^2 + y^2) dx - 5xy \, dy = 0, \, y(1) = 0 \), is:

• A. \(|x^2 - 4y^2|^5 = x^2\)
• B. \(|x^2 - 2y^2|^6 = x\)
• C. \(|x^2 - 4y^2|^6 = x\)
• D. \(|x^2 - 2y^2|^5 = x^2\)

Answer 1

The Correct Option is A

The differential equation \( (x^2 + y^2) \, dx - 5xy \, dy = 0 \) with the initial condition \( y(1) = 0 \) will be solved by following these steps:

The given differential equation is:

\((x^2 + y^2) \, dx - 5xy \, dy = 0\)

Rearranging yields:

\(\frac{dx}{dy} = \frac{5xy}{x^2 + y^2}\)

Using variable separation, we get:

\(\frac{x^2 + y^2}{5xy} \, dx = dy\)

Integrating both sides:

Left Side Integral: \( \int \frac{x^2 + y^2}{5xy} \, dx \)

Right Side Integral: \( \int dy \)

Considering the homogeneous nature of the equation and integrating leads to:

\(|x^2 - 4y^2| = C \cdot x^{\frac{2}{5}}\)

Applying the initial condition \( y = 0 \) when \( x = 1 \):

\(|1^2 - 4(0)^2| = C \cdot 1^{\frac{2}{5}}\)

This results in \( C = 1 \).

Substituting \( C = 1 \) back gives:

\(|x^2 - 4y^2|^5 = x^2\)

The solution is \(|x^2 - 4y^2|^5 = x^2\), which satisfies the differential equation and the initial condition.