ODE5.3

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xy''-y'=3x^{2}\,

Make this substitution to reduce the ODE to first order:

Let p=y'\,

xp'-p=3x^{2}\,

p'-{\frac  {1}{x}}p=3x\,

Let the integrating factor \rho =e^{{-{\frac  {1}{x}}\,dx}}=e^{{-\ln x}}={\frac  {1}{x}}\,

Multiply through by \rho \,.

{\frac  {1}{x}}p'-{\frac  {1}{x^{2}}}p=3\,

This is equivalent to

{\frac  {d}{dx}}\left[{\frac  {1}{x}}p\right]=3\,

{\frac  {1}{x}}p=3x+c_{1}\,

p=3x^{2}+c_{1}x\,

Since p=y'\,,

y'=3x^{2}+c_{1}x\,

y=x^{3}+{\frac  {c_{1}}{2}}x^{2}+c_{2}\,

Ordinary Differential Equations

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