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Solve x y''+2y'=6x

Short answer

Multiply by x to get exact derivative d/dx(x²y') = 6x²; integrate to obtain y' = 2x + c₁/x². General solution: y = x² − c₁/x + c₂.

Solve the differential equation x d²y/dx² + 2 dy/dx = 6x.

x y'' + 2y' = 6x

Method: exact derivative

Multiply both sides by x:

x²y'' + 2xy' = 6x²

The left side is d/dx (x²y'). So

d/dx (x² y') = 6x²

x²y' = 2x³ + c₁

y' = 2x + c₁/x²

Integrate again

y = x² − c₁/x + c₂

or, writing the constants as C₁ = −c₁ and C₂ = c₂: y = x² + C₁/x + C₂.

(This agrees with the Cauchy–Euler method: with z = log x, (θ² + θ)y = 6e^(2z), CF = c₁ + c₂e^(−z), PI = e^(2z) = x².)

✓ Answer checked by Claude Opus 5.5 (high effort)

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