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My question is, given the general solution $$Y=C_1x+c_2\dfrac{1}{x}$$ find the differential equation

My attempt:

I have derivated the equation and then look for the constants:

$$Y'=C_1+c_2\dfrac{-1}{x^2}$$ $$Y''=c_2\dfrac{2}{x^3}$$

so $c_2=y''\dfrac{x^3}{2}$
and substitute in $y'$: $$c_1=y'+\dfrac{y''x}{2}$$
then $$y=(y'+\dfrac{y''x}{2})x+(y''\dfrac{x^3}{2})(\dfrac{1}{x})$$

and the equation its suppose to be

$$y''x^2+y'x=0$$

but when i solve it, it doesn't check with the general solution given, so, can someone please tell me what I'm doing wrong?

Thanks.

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Given that $Y=C_1x+C_2\dfrac{1}{x}\implies xY=C_1 x^2+C_2$

Differentiating with respect to $x$,

$Y+xY'=2xC_1$ . . . . .$(1)$

Again differentiating with respect to $x$,

$Y'+Y'+xY''=2C_1\implies 2C_1=xY''+2Y'$

Putting the value of $2C_1$ in equation $(1)$ we have ,

$Y+xY'=x(xY''+2Y')\implies x^2Y''+xY'-Y=0$

This is the required differential equation.



If you want to cross-check the result, then take $$ x^2Y''+xY'-Y=0 \quad. . . . (2)$$

Putting $x=e^z\implies z=\log x$

Then $Y'=\frac{dY}{dx}=\frac{dY}{dz}\frac{dz}{dx}=\frac{1}{x}\frac{dY}{dz}\implies xY'=\frac{dY}{dz}=DY$(say), where $D\equiv \frac{d}{dz}$

similarly, $x^2Y''=D(D-1)Y$

Now from $(2)$,

$\{D(D-1)+D-1\}Y=0\implies (D^2-1)Y=0\implies Y=C_1 e^z+C_2e^{-z}\implies Y=C_1 x+C_2 \frac{1}{x}$

where $C_1$ and $C_2$ are arbitrary independent constants.

Hence the result holds.

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Note that if $y_h=c_1x+c_2/x$ implies that $y_1=x$ and $y_2=1/x$ are solutions of: $$y''+P(x)y'+Q(x)y=0$$ Then \begin{cases} (y_1)'' + P(x)(y_1)' + Q(x)y_1=0 \\ (y_2)'' + P(x)(y_2)' + Q(x)y_2=0\end{cases} \begin{cases} 0+ P(x) + Q(x)x=0\implies P(x)=-xQ(x) \\ \frac{2}{x^3} -P(x)\frac{1}{x^2}+ Q(x)\frac{1}{x}=0\end{cases}

\begin{cases} 0+ P(x) + Q(x)x=0\implies P(x)=-xQ(x) \\ \frac{2}{x^2} -(-xQ(x)\frac{1}{x})+ Q(x)=0\end{cases} Then $$Q(x)=\frac{-1}{x^2}\implies P(x)=\frac{1}{x}$$ So $$y''+P(x)y'+Q(x)y=0$$ $$y''+\frac{y'}{x}-\frac{y}{x^2}=0$$ $$x^2y''+xy'-y=0$$

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Note that $$ xy'=c_1x-\frac{c_2}x $$ which is the given solution up to a sign change. Thus repeating this combination of operations $$ x(xy')'=c_1x+\frac{c_2}x $$ returns the original solution. Thus the differential equation is $$ 0=x(xy')'-y=x^2y''+xy'-y. $$

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