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

• zenterix
zenterix
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

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:

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?

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.

zenterix
Substition
$$y=z e^{-x}$$
seems helpful. Have you tried it ?

zenterix
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 $$\frac{d^2}{dx^2} (e^{ax}g) = (g'' + 2ag' + a^2g)e^{ax}.$$

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.]

Last edited:

## What are the general methods used to solve the integrals ##\int\frac{e^{-x}}{x^2}dx## and ##\int \frac{e^{-x}}{x}dx##?

These integrals are typically solved using special functions or advanced techniques. For the integral ##\int\frac{e^{-x}}{x^2}dx##, one might use integration by parts or recognize it as related to the exponential integral function. The integral ##\int \frac{e^{-x}}{x}dx## is directly related to the Exponential Integral function, denoted as Ei(x).

## Can these integrals be expressed in terms of elementary functions?

No, these integrals cannot be expressed in terms of elementary functions. They are usually represented using special functions like the Exponential Integral function Ei(x).

## What is the Exponential Integral function, and how does it relate to these integrals?

The Exponential Integral function, denoted as Ei(x), is a special function that arises in problems involving integrals of the form ##\int \frac{e^t}{t}dt##. Specifically, ##\int \frac{e^{-x}}{x}dx## can be written in terms of the Exponential Integral function as ##-Ei(-x)##.

## What are the definite integrals of these functions from 0 to infinity?

The definite integral ##\int_0^\infty \frac{e^{-x}}{x^2}dx## converges to 1, while the integral ##\int_0^\infty \frac{e^{-x}}{x}dx## diverges because the integrand has a singularity at x = 0.

## Are there numerical methods to approximate these integrals?

Yes, numerical methods such as quadrature rules (e.g., Simpson's rule, trapezoidal rule) or adaptive algorithms can be used to approximate these integrals. Software packages like MATLAB, Mathematica, and numerical libraries in Python (e.g., SciPy) have built-in functions to compute these integrals numerically.

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