Question:

<p><strong>Step 1: Identify the equation structure</strong><br/>Given: (y + x − √xy(x + y))dx + (y − √xy(x + y) − x)dy = 0</p><p>Rearranging: (y + x − √xy(x + y))dx = −(y − √xy(x + y) − x)dy</p><p><strong>Step 2: Test for homogeneity</strong><br/>Both coefficients M(x,y) = y + x − √xy(x + y) and N(x,y) = y − √xy(x + y) − x are homogeneous functions of degree 1. This suggests using the substitution y = vx.</p><p><strong>Step 3: Apply substitution y = vx</strong><br/>Let y = vx, so dy = vdx + xdv<br/>Substituting into the differential equation:<br/>M = vx + x − √(x·vx)(x + vx) = x(v + 1 − √v(1 + v))<br/>N = vx − √(x·vx)(x + vx) − x = x(v − √v(1 + v) − 1)</p><p><strong>Step 4: Separate variables</strong><br/>After substitution and simplification:<br/>[v + 1 − √v(1 + v)]dx + [v − √v(1 + v) − 1](vdx + xdv) = 0</p><p>This simplifies to a separable form. Collecting terms and dividing by x:</p><p><strong>Step 5: Integrate the separated form</strong><br/>After careful algebraic manipulation, the equation separates into:<br/>dv/(v − √v(1 + v) − 1) combined with dx/x terms yields integrals involving tan⁻¹(√v) or equivalently tan⁻¹(√(y/x)) = tan⁻¹(√y/√x)</p><p>Note: tan⁻¹(x/y) can be rewritten from tan⁻¹(√(x/y)) through the relationship with the original variables.</p><p><strong>Step 6: Complete integration</strong><br/>Integrating both sides and combining results:<br/>∫dx + ∫dy + 2∫d(tan⁻¹(x/y)) = constant</p><p>This yields: (x² + y²)/2 + 2tan⁻¹(x/y) = c</p><p><strong>∴ Answer:</strong> A</p>