Transformation of a Cauchy-Euler equation

In summary, the conversation discussed the process of transforming a Cauchy-Euler equation into an equation with constant coefficients. This was done by making the substitution x = e^t, which led to an inhomogeneous 2nd order ODE. The standard method of finding complementary solutions and a particular solution was then used to solve the equation. The final solution was found to be y= -(ln(x)+1)*ln(x)/4 +A*x² +B, where A and B are constants.
  • #1
jawhnay
37
0
Can anyone explain to me how I would go about transforming a Cauchy-Euler equation for an equation such as:
x2y'' - xy' = ln x

I know you have to start with x = et or t = ln x however I'm not sure what to do next...
 
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  • #2
Well making the substitution [tex]x=e^t[/tex] does indeed get the solution, you should get via chain rule:

[tex]\frac{d}{dx}=e^{-t}\frac{d}{dt}, \frac{d^2}{dx^2}=e^{-2t}(\frac{d^2}{dt^2}-\frac{d}{dt})[/tex]

which when substituted into your original equation should yield an inhomogeneous 2nd order ODE with constant coefficients which we can do via the standard method of finding complementary solutions and then a particular solution.
 
  • #3
With z(x)=y'(x), the ODE is of the first order :
x²z'+xz=ln(x)
Solving x²z'-xz=0 leads to z=c*x where c=constant
Then bing back z=u(x)*x into x²z'-xz=ln(x)
which leads to u' =ln(x)/(x^3)
u(x)=-(2*ln(x)+1)/(4*x²)+C where C=constant
y'=z(x)=u(x)*x=-(2ln(x)+1)/(4*x)+C*x
y= -(ln(x)+1)*ln(x)/4 +A*x² +B where A, B are constants.
 
  • #5


To transform a Cauchy-Euler equation, you need to make a substitution that will turn the equation into a standard form that can be easily solved. In this case, since the equation involves a natural logarithm, it would be helpful to substitute x = e^t. This will transform the equation into one that involves only t and its derivatives.

So, let's start by making the substitution x = e^t. This means that dx/dt = e^t and d^2x/dt^2 = e^t. Substituting these into the original equation, we get:

(e^t)^2y'' - e^t y' = ln(e^t)

Simplifying, we get:

e^2t y'' - e^t y' = t

Now, we can divide both sides by e^t to get the equation in standard form:

y'' - y' = t/e^t

This is now a standard form for a Cauchy-Euler equation, which can be solved using various methods such as the method of undetermined coefficients or the method of variation of parameters.

I hope this helps to explain the process of transforming a Cauchy-Euler equation. Keep in mind that the specific substitution you make may vary depending on the form of the equation, so it's important to carefully analyze the equation and choose a substitution that will simplify it as much as possible.
 

1. What is a Cauchy-Euler equation?

A Cauchy-Euler equation is a type of differential equation that is expressed in the form: anxn + an-1xn-1 + ... + a1x + a0 = 0, where n is a positive integer and the coefficients an, ..., a1, a0 are constants.

2. Why is it important to transform a Cauchy-Euler equation?

Transforming a Cauchy-Euler equation can make it easier to solve, as it can be converted into a simpler form. This transformation also allows us to find a general solution that can be applied to a wider range of problems.

3. What are the steps involved in transforming a Cauchy-Euler equation?

The steps involved in transforming a Cauchy-Euler equation are:

  1. Let x = et, and rewrite the equation in terms of t
  2. Use the chain rule to find the derivatives of x with respect to t
  3. Substitute the derivatives into the equation and simplify
  4. Use the substitution u = tn to simplify the equation
  5. Solve for u and substitute back in for x to find the general solution

4. Can Cauchy-Euler equations be transformed into any other form?

Yes, Cauchy-Euler equations can be transformed into different forms depending on the specific problem. Some common forms include the power series form and the Laplace transform form.

5. How are Cauchy-Euler equations used in real-world applications?

Cauchy-Euler equations can be used to model various physical phenomena, such as the motion of a spring or the growth of a population. They are also commonly used in engineering and physics to solve problems involving differential equations.

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