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I need to find a differential equation that satisfies the solutions $$y_1 (x) = x $$ and $$ y_2 (x) = \ln(x)$$ on the interval $( 0, + \infty)$. Since these two solutions are linearly independent, I know that the differential equation will have to be of second order. But I cannot come up with one. Is there some general method to come up with differential equations, given some solutions?

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A general solution (if you want a linear differential equation) is

$$y=c_1x + c_2 \ln{x}$$ then $$y'=c_1+\frac{c_2}{x}$$ and $$y''=\frac{-c_2}{x^2}$$ One way to come up with a differential equation is to use these three equations and eliminate the constants $c_1$ and $c_2$. From the last equation you know that $c_2=-x^2y''$. Putting this in the second equation yields $$y'=c_1-xy''$$ or $$c_1=y'+xy''$$ which you can insert into the first equation for $c_1$ to yield $$y= (y'+xy'')x-x^2y''\ln{x}$$

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    $\begingroup$ Thank you! That was very clear and helpful. $\endgroup$
    – Kamil
    Jan 12, 2016 at 14:17
  • $\begingroup$ More generally, $$\left|\begin{array}{cc}y''_1&y''_2\\ y'_1&y'_2\end{array}\right|\cdot y=\left|\begin{array}{cc}y''&y''_2\\ y'&y'_2\end{array}\right|\cdot y_1+\left|\begin{array}{cc}y''_1&y''\\ y'_1&y'\end{array}\right|\cdot y_2,$$ which is equivalent to $$\left|\begin{array}{cc}y''_1&y''_2\\ y'_1&y'_2\end{array}\right|\cdot y=\left|\begin{array}{cc}y_1&y_2\\ y'_1&y'_2\end{array}\right|\cdot y''+\left|\begin{array}{cc}y''_1&y''_2\\ y_1&y_2\end{array}\right|\cdot y'.$$ $\endgroup$
    – Did
    Jan 13, 2016 at 10:12

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