 Repeated-root constant-coefficient differential equation

ID: repeated-root-constant-coefficient-differential-equation

The homogeneous equation has linearly independent solutions

e^{r x}

and

x e^{r x}

, with nonzero Wronskian

e^{2 r x}

. Setting

y = e^{r x} u

converts the forced equation to

u^{''} = e^{- r x} f

f = e^{r x}

, the particular response is

x^{2} e^{r x} /2

. This substitution explains the extra powers of

x

required by repeated-root resonance.

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