Question

The solution of the differential equation
$ xy 𝑑𝑥 − (x^2 + y^2) 𝑑y = 0, y(0) = 1$ is

• A. $𝑦= e^{\frac{x}{y}}$
• B. $y^2= e^{\frac{x^2}{y^2}}$
• C. $y^2= e^{\frac{x}{y}}$
• D. $y= e^{\frac{x^2}{y^2}}$

Answer 1

The Correct Option is B

Solving the Given Differential Equation

Step 1: Rewriting the Differential Equation 

The given differential equation can be written as:

xy dx = (x² + y²) dy

Dividing both sides by xy², we get:

(dx / y) = ((x² + y²) / xy²) dy

Step 2: Simplifying

Rewriting the equation:

(dx / y) = (x / y²) dy + (1 / x) dy

Step 3: Substituting Variables

Let u = x / y, then x = uy and dx = u dy + y du.

Substituting these into the equation gives:

(u dy + y du) / y = (u² + 1) / u dy

Step 4: Further Simplification

This simplifies to:

u dy + du = (u² + 1) / u dy

Rearranging terms and integrating:

du = ((u² + 1) / u - u) dy = (1 / u) dy

Step 5: Integration

Integrating both sides:

∫ du = ∫ (1 / u) dy

which results in:

u = ln(y) + C

Step 6: Applying the Initial Condition

Given that y(0) = 1, solve for C. Substituting u = x / y back:

x / y = ln(y) + C

Step 7: Final Expression

Squaring both sides to eliminate the logarithm leads to:

y² = e^x² y²