Differential equations

A first-order linear differential equation, y' + p(x)y = g(x), cannot simply be separated like the simplest exponential-growth equations: instead, multiplying through by an integrating factor mu(x) = e to the integral of p(x)dx makes the left side collapse into the derivative of a single product. For y' + (2/x)y = x on an interval where x > 0, the integrating factor is x^2, turning the equation into (x^2 times y)' = x^3, which integrates to x^2 times y = x^4/4 + C, giving y = x^2/4 + C/x^2.

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