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The solution of the differential equation x dy/dx + 2y = x² (x ≠ 0) with y(1) = 1, is:
- y = 4/5 x³ + 1/(5x²)
- y = x³/5 + 1/(5x²)
- y = x²/4 + 3/(4x²)
- y = 3/5 x² + 1/(4x²)
Correct answer: y = x²/4 + 3/(4x²)
Solution
Rewrite as dy/dx+2y/x=x; integrating factor is x^2, so d/dx(x^2 y)=x^3, giving x^2 y=x^4/4+C, i.e. y=x^2/4+C/x^2. Using y(1)=1, C=3/4, so y=x^2/4+3/(4x^2).
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