anonymous
  • anonymous
How do I find two linear solutions for the linear equation: y" + (1/x)y' = 0 ?
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
Hi hollyn, If you make a substitution\[v=y'\]then\[v'=y''\]and your equation becomes\[v'+\frac{1}{x}v=0\]This equation is separable,\[\frac{dv}{dx}=-\frac{v}{x} \rightarrow \frac{dv}{v}=-\frac{dx}{x}\]Integrate both sides\[\ln v = - \ln x +c \]where c is some constant.Exponentiating both sides leaves you with\[v=\frac{c_1}{x}\]where c_1 is some constant. But remember, \[v=y'\]so\[\frac{dy}{dx}=\frac{c_1}{x} \rightarrow y=c_1 \ln x + c_2\]where c_2 is an arbitrary constant.

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