If y=y(x) is the solution of the differential equation xdy/dx + 2y = x² satisfying y(1) = 1, then y(1/2) is equal to

dy/dx + (2/x)y = x ⇒ I.F. = x²
∴ yx² = x⁴/4 + C
As, y(1) = 1
1(1)² = 1⁴/4 + C
1 = 1/4 + C
C = 3/4
∴ yx² = x⁴/4 + 3/4
y = x²/4 + 3/(4x²)
y(x = 1/2) = (1/2)²/4 + 3/(4(1/2)²) = 1/16 + 3/4 = 13/16
This solution doesn't match the given options. Let's re-examine the solution.
The differential equation is x(dy/dx) + 2y = x². This is a linear differential equation of the form dy/dx + P(x)y = Q(x), where P(x) = 2/x and Q(x) = x.
The integrating factor (IF) is given by e^(∫P(x)dx) = e^(∫(2/x)dx) = e^(2ln|x|) = x². Multiplying the differential equation by the integrating factor, we get:
x²(dy/dx) + 2x²y/x = x³
x²(dy/dx) + 2xy = x³
d/dx(x²y) = x³
Integrating both sides with respect to x:
∫d/dx(x²y)dx = ∫x³dx
x²y = x⁴/4 + C
Using the initial condition y(1) = 1:
1² * 1 = 1⁴/4 + C
1 = 1/4 + C
C = 3/4
Therefore, the solution is x²y = x⁴/4 + 3/4.
y = x²/4 + 3/(4x²)
Now, let's find y(1/2):
y(1/2) = (1/2)²/4 + 3/(4*(1/2)²) = 1/16 + 3/4 = 13/16
This is still not matching any of the given options. There may be an error in the problem statement or the options provided. The calculated value is 13/16