Separable equations

Question

Find the solution of the differential equation that satisfies the given initial condition.

x ln(x) = y(1 +  √(15 + y²)y',    y(1) = 1 .

Answer 1

Separate variables and integrate to get ∫xlnxdx = ∫[y(1 + √(15+y²)]dy

Evaluate the left hand side by parts: (1/2)x²lnx - (1/4)x² = ∫ ydy + ∫y√(15+y²)dy

(1/2)x²lnx - (1/4)x² = (1/2)y² + (1/2)∫u^1/2du, where u = 15+y²

(1/2)x²lnx - (1/4)x² = (1/2)y² + (1/3)(15+y²)^3/2 + C

Use the fact that (1,1) is on the curve, to find C.