5A (Differential Equations)

Let

I (x) = exp (\int^{x} p (s) d s) .

(I y)^{'} = I f

gives

y (x) = I (x)^{- 1} (C + \int^{x} I (t) f (t) d t) .

y_{1}, y_{2}

are distinct solutions, their difference is a nonzero homogeneous solution, so every solution is

y = y_{1} + C (y_{2} - y_{1}) .

b

For

x > 0

x

gives

y^{'} - (2 x + x^{- 1}) y = x .

e^{- x^{2}} / x

, so

y = x e^{x^{2}} (C + \int_{0}^{x} e^{- t^{2}} d t) .

Every choice except

C = - \int_{0}^{\infty} e^{- t^{2}} d t = - \frac{π}{2}

grows like

x e^{x^{2}}

y_{b} (x) = - x e^{x^{2}} \int_{x}^{\infty} e^{- t^{2}} d t .

Integration by parts, or the Gaussian-tail asymptotic, gives

\int_{x}^{\infty} e^{- t^{2}} d t \sim \frac{e ^{- x^{2}}}{2 x}, y_{b} (x) \to - \frac{1}{2} .

It starts from

0

x = 0

, remains negative, and decreases monotonically toward the horizontal asymptote

- 1/2

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