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$$xy'' + 2y' - xy = e^x$$

is the equation. Here's what I did:

Divide by x

First, solve the homogenous equation.

$$y'' + \frac{2}{x}y' - y =0$$

Perform a substitution $y=u(x) \cdot z(x)$, where $u(x) = e^{-\frac{1}{2}\int{p(x)dx}}$

In this case, $p(x) = \frac{2}{x}$, and $q(x) = -1$

Find $u', u''$

After plugging in the necessary values, we get an equation: $$z'' - z = 0$$, the solution of which is $$C_1e^x + C_2e^{-x}$$

We plug this in $y=u\cdot z$ and get:

$$y=C_1 \frac{e^x}{x} + C_2 \frac{e^{-x}}{x}$$

I am fairly sure I made no mistakes up until this point. Next, I tried to solve the equation by variation of constants, where I get the Wronskian $-\frac{2}{x^2}$.

Thus, $C_1' = \frac{1}{2}$, so $C_1 = \frac{x}{2} + C_3$

And $C_2' = -\frac{e^{2x}}{2}$, so $C_2 = -\frac{e^{2x}}{4} + C_4$

When I plug in these values into $y$, I get

$$y=\frac{e^x}{2} + C_3 \frac{e^x}{x} - \frac{e^x}{4x} + C_4\frac{e^{-x}}{x}$$

Now, when I check the solution on WolframAlpha, the third term in my solution is extra. Can anyone spot my mistake because I am unable to do so? I have been trying for at least several hours.

NOTE: It is imperative for this exercise to be done with variation of constants.

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    $\begingroup$ $C_3 \frac{e^x}{x} - \frac{e^x}{4x}=C_5 \frac{e^x}{x}$ $\endgroup$ May 24, 2021 at 9:11
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    $\begingroup$ Looking at all your wrote,, making from the start $y(x)=\frac{e^x}x z(x)$ gives the simple $z''+2z'=1$. Reduction of order gives $z'=\frac 12+c_1 e^{-2x}$ then $z=\frac x 2+c_1 e^{-2x}+c_2$. Multiply by $\frac{e^x}x$ and expand. In any manner, you cannot have more than two arbitrary constants. $\endgroup$ May 24, 2021 at 9:24
  • $\begingroup$ @ClaudeLeibovici I do not have more than two constants. I couldn't name the new constants C1 and C2 because I used that notation already. I could have called them $K_1$ and $K_2$ though. $\endgroup$
    – john doe
    May 24, 2021 at 9:30
  • $\begingroup$ And isn't $u(x) = \frac{1}{x} $ ? $\endgroup$
    – john doe
    May 24, 2021 at 9:31
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    $\begingroup$ You did not make anything wrong. My first comment was just $Cte+Cte=Cte$. My second comment was to show that, starting from your work, we could to it a bit faster. That's all. $\endgroup$ May 24, 2021 at 9:56

3 Answers 3

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One could also simply observe that from the general Leibniz rule for product differentiation $$xy''+2y'=(xy)''$$ and compute the particular solution via undetermined coefficients, $$ z''-z=e^x,~~~ z_p=Axe^x\implies 2A=1 \\~\\ z(x)=xy(x)=\frac12xe^x+C_1e^x+C_2e^{-x} $$

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$$xy'' + 2y' - xy = e^x$$ $$(xy'+y)'-xy=e^x$$ $$(xy)''-xy=e^x$$ $$(xy)''-(xy)'+(xy)'-xy=e^x$$ $$(e^{-x}(xy)')'+(xye^{-x})'=1$$ $$(e^{-x}(xy)'+xye^{-x})'=1$$ $$(e^{-2x}(xye^{x})')'=1$$ Integrate. $$\boxed {xy=c_1e^x+c_2e^{-x}+\dfrac 12 xe^x}$$

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First the homogeneous solution is $y_h = \frac{c_1 e^{-x}}{x}+\frac{c_2 e^x}{x}$

Second, making the particular as $y_p = \frac{c_1(x) e^{-x}}{x}+\frac{c_2(x) e^x}{x}$ after substitution into the complete ODE we have

$$ e^{-x} c_1''(x)-2 e^{-x} c_1'(x)+e^x c_2''(x)+2 e^x c_2'(x)-e^x=0 $$

due to de independence between $c_1$ and $c_2$ we can arrange this as

$$ \cases{ e^{-x} c_1''(x)-2 e^{-x} c_1'(x)-e^x = 0\\ e^x c_2''(x)+2 e^x c_2'(x) = 0 } $$

making $u_1 = c_1',\ \ u_2 = c_2'$ we follow with

$$ \cases{ e^{-x} u_1'(x)-2 e^{-x} u_1(x)-e^x = 0\\ e^x u_2'(x)+2 e^x u_2(x) = 0 } $$

with particular solutions

$$ \cases{ u_1(x) = x e^{2 x}\\ u_2(x) = 0 } $$

or integrating particularly

$$ \cases{ c_1(x) = \frac{1}{4} e^{2 x} (2 x-1)\\ c_2(x) = 0 } $$

then finally

$$ y(x) = y_h+y_p = \frac{c_1 e^{-x}}{x}+\frac{c_2 e^x}{x}+\frac{e^x \left(2 x-1\right)}{4 x} $$

or

$$ y(x) = \frac{c_1 e^{-x}}{x}+\left(\frac{c_2 e^x}{x}-\frac{e^x}{4x}\right)+\frac{x e^x}{2} $$

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