Solve ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##

zenterix
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Homework Statement
Find the general solution to the differential equation
Relevant Equations
##y''+2y'+y=\frac{e^{-x}}{x^2}##
The characteristic equation has a zero discriminant and the sole root of ##-1##.

The general solution to the associated homogeneous equation is thus

$$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$

Now we only need to find one particular solution of the non-homogeneous equation.

The righthand side of the non-homogeneous equation, call it ##R(x)##, is not a polynomial, nor does it fall in the category of ##p(x)e^{mx}## with ##p## being a polynomial. These are perhaps the two easiest special cases of ##R(x)## to solve because they can be solved by the method of undetermined coefficients.

##R(x)## is also not an exponential multiplied by an expression containing ##\sin## and/or ##\cos##, which can also be solved by the method of undetermined coefficients.

Thus, I have chosen to solve using a more general method based on the following theorem in Apostol's Calculus, Volume I

1699674679317.png


We have ##v_1(x)=1## and ##v_2(x)=x## as our solutions to ##L(y)=0##, ie the homogeneous equation.

The Wronskian of ##v_1## and ##v_2## is

$$W(x)=
\begin{vmatrix}
1 & 0\\
x & 1
\end{vmatrix}=1
\tag{2}$$

Thus,

$$t_1(x)=-\int x\cdot\frac{e^{-x}}{x^2}dx=\int\frac{e^{-x}}{x}dx\tag{3}$$

$$t_2(x)=\int \frac{e^{-x}}{x^2}dx\tag{4}$$

How do we solve these integrals?

Math software doesn't get me very far:

1699675123903.png


The window on the bottom is a sort of "math tutor" in Maple that shows all the steps in the calculation of the integral. However, for some integrals this step-by-step isn't available.

Now ##i_1(x)## is probably some variable defined within the depths of the dungeons of Maple documentation.

Still, what technique for solving integrals works for these integrals?
 
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In that case, here is a follow up question.

The solution to the problem in the back of the book is

$$y=(c_1+c_2x-\log{|x|})e^{-x}$$

which seems to imply that the particular solution they are using for the non-homogeneous equation is

$$y_p(x)=-e^{-x}\log{|x|}$$

How did they arrive at this particular solution?
 
zenterix said:
$$y_p(x)=-e^{-x}\log{|x|}$$

How did they arrive at this particular solution?
I would look for a function of the form ##y_p(x) = e^{-x}f(x)##. Even without seeing the solution, that looks like a good idea.
 
Substition
y=z e^{-x}
seems helpful. Have you tried it ?
 
PeroK said:
I would look for a function of the form ##y_p(x) = e^{-x}f(x)##. Even without seeing the solution, that looks like a good idea.
Indeed it is a good idea and the answer comes out quite easily.
 
zenterix said:
Homework Statement: Find the general solution to the differential equation
Relevant Equations: ##y''+2y'+y=\frac{e^{-x}}{x^2}##

Consider the identity <br /> \frac{d^2}{dx^2} (e^{ax}g) = (g&#039;&#039; + 2ag&#039; + a^2g)e^{ax}.
 
zenterix said:
The general solution to the associated homogeneous equation is thus

$$y_h(x)=e^{-x}(c_1+c_2x)\tag{1}$$

zenterix said:
We have ##v_1(x)=1## and ##v_2(x)=x## as our solutions to ##L(y)=0##, ie the homogeneous equation.

Shouldn't that be ##v_1(x) =e^{-x}## and ##v_2(x) = x e^{-x}##?

The method of solution given by the theorem in Apostol will work out nicely. [Edited to add this comment.]
 
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